ijpAL  PHILOSOPHY. 

BY  PROFESSOR  EVERETT. 


PART  IV. 
SOUND    AND     LIGHT 


THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


u 


/v 


ELEMENTARY    TREATISE 


OX 


NATURAL  PHILOSOPHY 


BY 

A.  PEIVAT   DESCHANEL, 

FORMERLY  PROFESSOR  OP  PHYSICS   IN  .THE  LYC^E  LOUIS-LE -GRAND, 
INSPECTOR  OF  THE  ACADEMY  OF  PARIS. 


TRANSLATED    AND    EDITED,   WITH    EXTENSIVE    MODIFICATIONS 

BY  J.  D.  EVEEETT,  M.  A.,  D.  0.  L.,  F.  R.  S.,  F.  R.  8.  E., 

PROFESSOR    OF    NATURAL    PHILOSOPHY    IN    THE    QUEEN'S    COLLEGE,  BELFAST. 


IN     FOUR     PARTS. 


PART    IV. 

SOUND     AND     LIGHT. 

ILLUSTRATED    BY 
193     ENGRAVINGS     ON     WOOD,     AND     ONE     COLORED     PLATE. 


TENTH  EDITION. 


NEW  YORK: 
D.  APPLETON  AND  COMPANY, 

1,   8,   AND   5   BOND   STREET. 

1891. 


PREFACE  TO   THE  TENTH  EDITION  OF 
PART    IV. 


THIS  Part,  even  in  its  original  form,  contained  large  portions  which  were  re- 
written rather  than  translated,  and  the  last  two  chapters  had  no  place  at  all 
in  the  original  French.  In  the  sixth  edition  the  numbering  of  the  chapters 
and  sections  was  altered,  to  make  it  consecutive  with  the  other  three  parts, 
;m;l  additional  matter  was  introduced  under  several  heads.  In  the  seventh 
edition  two  pages  on  Concave  Diffraction  Gratings  were  added.  The  tenth 
edition  contains  additions  relating  to  beats,  recent  measures  of  the  velocity  of 
light,  goniometers,  convex  mirrors,  and  nodal  points  of  lenses. 

BELFAST,  March,  1838. 


NOTE   PREFIXED   TO   FIRST  EDITION. 

IN  the  present  Part,  the  chapters  relating  to  Consonance  and  Dissonance, 
Colour,  the  Undulatory  Theory,  and  Polarization,  are  the  work  of  the  Editor; 
besides  numerous  changes  and  additions  in  other  places. 

The  numbering  of  the  original  sections  has  been  preserved  only  to  the  end 
of  Chapter  LX. ;  the  two  last  chapters  of  the  original  having  been  transposed 
for  greater  convenience  of  treatment.  With  this  exception,  the  announcements 
made  in  the  "Translator's  Preface,"  at  the  beginning  of  Part  I.,  are  applicable 
to  the  entire  work. 


1109429 


CONTENTS-PART  IV. 


THE  NUMBEUS   REFEE  TO   THE   SECTION'S. 


ACOUSTICS. 

CHAPTER  LXII.    PRODUCTION  AND  PROPAGATION  OF  SOUND. 

Sound  results  from  vibratory  movement.  Examples  and  definitions,  866.  Musical  sound. 
867.  Vehicle  of  sound,  868.  Mode  of  propagation.  Relation  between  period,  wave- 
length, and  velocity  of  propagation,  869.  Undulation  considered  geometrically ;  For- 
ward velocity  of  particle  proportional  to  its  condensation,  870.  Propagation  in  open 
space.  Inverse  squares.  Propagation  in  tubes.  Energy  of  undulations,  871.  Dis- 
sipation; Conversion  into  heat,  872.  Velocity  of  sound  in  air;  Mode  of  observing,  and 
results,  873.  Theoretical  computation,  874-876.  Newton  and  Laplace,  877.  Velo- 
city in  gases,  878.  In  liquids.  Colladon's  experiment ;  Theoretical  computation,  879. 
In  solids.  Biot's  experiment.  Wertheim's  results,  880.  Theoretical  computation, 
881.  Reflection  of  sound,  and  Sondhaus'  experiment  on  refraction  of  sound,  882,  883. 
Echo,  884.  Speaking  and  hearing  trumpets,  885.  Interference  of  sounds,  886.  In- 
terference of  direct  and  reflected  waves.  Nodes  and  antinodes,  887.  Beats,  888, 

pp.  865-893. 

Note  A.     Rankine's  investigation, pp.  893,  894. 

Note  B.     Usual  investigation  of  velocity  of  sound, p.  894. 

Note  C.     Analysis  of  stationary  undulation, pp.  894,  895. 

Note  D.      Investigation  of  beats, p.   895. 

CHAPTEB  LXIII.    NUMERICAL  EVALUATION  OF  SOUND. 

Loudness,  pitch,  and  character.  Pitch  depends  on  frequency,  889.  Musical  intervals,  890. 
Gamut,  891.  Temperament.  Absolute  pitch,  892.  Limits  of  pitch  in  music,  893. 
Minor  and  Pythagorean  scales,  894.  Methods  of  counting  vibrations.  Siren,  895. 
Vibroscope  and  Phonautograph,  896.  Tonometer,  897.  Pitch  modified  by  approach 
or  recess,  898 .pp.  896-908. 

CHAPTER  LXIV.    MODES  OF  VIBRATION. 

Longitudinal  and  transverse  vibrations,  899.  Transverse  vibrations  of  strings,  900.  Their 
laws,  901.  Sonometer,  902.  Harmonics;  Segmental  vibration  of  strings,  903. 
Sympathetic  vibrations  or  resonance;  Sounding-boards,  904.  Longitudinal  vibra- 
tions of  strings,  905.  Stringed  instruments,  906.  Transversal  vibrations  of  solids; 
Chladni's  figures ;  Bells,  907.  Tuning-fork,  908.  Law  of  linear  dimensions,  909. 
Organ-pipes,  and  experimental  organ,  910-912.  Bernouilli's  laws  for  overtones  of 
pipes,  913.  Position  of  nodes  and  antinodes,  914.  Explanation,  915.  Analogous 
laws  for  rods  and  strings,  916.  Application  to  measurement  of  velocity  of  sound  in 
various  substances,  917.  Reed-pipes,  918.  Wind-instruments,  919.  Manometric 
flames,  920 pp.  908-926. 


Vi  TABLE   OF   CONTENTS. 

CHAPTEB  LXV.    ANALYSIS  OF  VIBRATIONS.     CONSTITUTION  OF 
SOUNDS. 

Optical  examination  of  sonorous  vibrations,  921.  Lissajous'  experiment.  Equations  of 
Lissajous'  curves,  922.  Optical  tuning,  923.  Other  modes  of  obtaining  Lissajous' 
curves.  Kaleidophone  and  Blackburn's  pendulum,  924.  Character  or  timbre.  Every 
periodic  vibration  consists  of  a  fundamental  simple  vibration  and  its  harmonics ;  and 
every  musical  note  consists  of  a  fundamental  tone  and  its  harmonics,  925.  Helm- 
holtz's  resonators,  926.  Vowel-sounds,  927.  Phonograph,  928, .  .  pp.  927-940. 

CHAPTER  LXVI.     CONSONANCE,  DISSONANCE,  AND  RESULTANT 
TONES. 

Concord  and  discord,  929.  Helmholtz's  theory,  930.  Beats  of  harmonics,  931.  Beating 
notes  must  be  near  in  pitch,  932.  Imperfect  concord,  933.  Resultant  tones,  934,  935, 

pp.  941-946. 


OPTICS. 

CHAPTEB  LXVII.    PROPAGATION  OF  LIGHT. 

Light.  Hypothesis  of  scther,  936.  Excessive  frequency  of  vibration.  Sharpness  of 
shadows  due  to  shortness  of  waves,  937.  Images  produced  by  small  apertures,  938. 
Shadows.  Umbra  and  penumbra,  939.  Velocity  of  light,  940.  Fizeau's  experiment 
and  Cornu's,  941.  Foucault's  experiment,  942.  Later  determinations  of  velocity  by 
Foucault's  method,  943.  Eclipses  of  Jupiter's  satellites,  944.  Aberration,  945. 
Photometry,  946.  Bouguer's  photometer,  947.  Rumford's,  948.  Foucault's,  949. 
Bunsen's  and  Letheby's,  950.  Photometers  for  very  powerful  lights,  951. 

pp.  947-966. 

CHAPTER  LXVIIL    REFLECTION  OF  LIGHT. 

Reflection,  952.  Its  laws,  953.  Artificial  horizon,  954.  Irregular  reflection,  955.  Mir- 
rors, 956.  Plane  mirrors,  957.  Images  of  images,  958.  Parallel  mirrors,  959. 
Mirrors  at  right  angles,  960.  Mirrors  at  60°,  961.  Kaleidoscope,  962.  Pepper's 
ghost,  963.  Deviation  produced  by  rotation  of  mirror,  964.  Sextant,  965.  Spheri- 
cal mirrors,  966.  Conjugate  foci,  967.  Principal  focus,  968.  March  of  conjugate 
foci,  969.  Construction  for  image,  970.  Size  of  image,  971.  Phantom  bouquet,  972. 
Image  on  screen,  973.  Caustics,  primary  and  secondary  foci,  974.  Two  focal  lines, 
975.  Virtual  images,  976,  977.  Convex  mirrors,  978.  Cylindric  mirrors.  Anamor- 
phosis, 979.  Ophthalmoscope  and  Laryngoscope,  980, pp.  967-991. 

CHAPTER  LXIX.    REFRACTION. 

Refraction,  981,  982.  Its  laws,  983.  Apparatus  for  verification,  984,  985.  Indices  of 
refraction,  986.  Critical  angle  and  total  reflection,  987.  Camera  lucida,  988.  Image 
by  refraction  at  plane  surface  when  incidence  is  normal,  989.  Caustic.  Position  of 
image  for  oblique  incidence,  990.  Refraction  through  plate,  991.  Multiple  imagas, 
992.  Superposed  plates ;  Astronomical  refraction,  993.  Refraction  through  prism, 
994.  Formulae,  995.  Construction  for  deviation ;  Minimum  deviation,  996.  Con- 
jugate foci  for  minimum  deviation,  997.  Double  refraction;  Iceland-spar,  998,  999, 

pp.  992-1012. 


TABLE   OF   CONTENTS. 


CHAPTER  LXX.     LENSES. 
\ 

Forms  of  lenses,  1000.  Principal  focus,  1001.  Optical  centre,  1002.  Conjugate  foci. 
Image  erect  or  inverted,  enlarged  or  diminished,  1003.  Formulae,  1004.  Conjugate 
foci  on  secondary  axis,  1005.  March  of  conjugate  foci,  1006.  Construction  for  image, 
1007.  Size  of  image,  1008.  Example,  1009.  Image  on  cross-wires,  1010.  Aberra- 
tion of  lenses,  1011.  Virtual  images,  1012.  Concave  lens,  1013.  Focometer,  1014. 
Refraction  at  a  single  spherical  surface,  1015.  Refraction  through  a  sphere,  1016. 
Camera  obscura,  1017.  Photographic  camera  and  photography,  1018.  Projection; 
Solar  microscope  and  magic  lantern,  1019,  1020 pp.  1013-1030. 

CHAPTER  LXXI.    VISION  AND  OPTICAL  INSTRUMENTS. 

Construction  of  eye,  1021.  Its  optical  working,  1022.  Adaptation  to  distance,  1023. 
Binocular  vision.  Data  for  judgment  of  distance,  1024.  Stereoscope,  1025.  Visual 
angle;  Magnifying  power,  1026.  Spectacles,  1027.  Magnification  by  a  lens.  Sim- 
ple microscope,  1028.  Compound  microscope  and  its  magnifying  power,  1029. 
Astronomical  telescope  and  its  magnifying  power;  Finder,  1030.  Place  for  eye; 
Bright  spot  and  its  relation  to  magnifying  power,  1031.  Terrestrial  eye-piece,  1032, 
Galilean  telescope.  Its  peculiarities.  Opera-glass,  1033.  Reflecting  telescopes, 
1034.  Silvered  specula,  1035.  Measure  of  brightness;  intrinsic  and  effective,  1036. 
Surfaces  are  equally  bright  at  all  distances.  Image  formed  by  theoretically  perfect 
lens  has  same  intrinsic  brightness  as  object;  but  effective  brightness  may  be  less. 
Same  principle  applies  to  mirrors.  Reason  why  high  magnification  often  produces 
loss  of  effective  brightness,  1037.  Intrinsic  brightness  of  image  in  theoretically 
perfect  telescope  is  equal  to  brightness  of  object.  Effective  brightness  is  the  same  if 
magnifying  power  does  not  exceed  — '  and  is  less  for  higher  powers,  1038.  Light 
received  from  a  star  increases  with  power  of  eye-piece  till  magnifying  power 
is  -»  1039.  Illumination  of  image  on  screen  is  proportional  to  solid  angle  subtended 
by  lens.  Appearance  .presented  to  eye  at  focus,  1040.  Field  of  view  in  astronomical 
telescope,  1041.  Cross-wires,  and  their  adjustment  for  preventing  parallax,  1042. 
Line  of  collimation,  and  its  adjustment,  1043.  Micrometers,  1044,  pp.  1031-1058. 

CHAPTER  LXXIL    DISPERSION.    STUDY  OF  SPECTRA. 

Analysis  of  colours  by  prism,  1045,  1046.  Newton's  method  of  obtaining  the  solar 
spectrum,  1047.  Modes  of  obtaining  a  pure  spectrum  either  virtual  or  real,  1048. 
Fraunhofer's  lines,  1049.  Invisible  ends  of  spectrum,  1050.  Phosphorescence  and 
fluorescence,  1051.  Decomposition  of  white  light,  1052.  Spectroscope,  1053.  Use 
of  collimator,  1054.  Classes  of  spectra,  1055.  Spectrum  analysis,  1056.  Inferences 
from  dark  lines  in  solar  spectrum,  1057.  Observations  of  chromosphere.  Spectra  of 
nebulae,  1058.  Doppler's  principle,  1059.  Spectra  of  artificial  lights.  Bodies 
illuminated  by  monochromatic  light,  1060.  Brightness  and  purity  of  spectra,  1061. 
Chromatic  aberration,  1062.  Achromatism.  "Dispersive  power,"  106*3,  1064. 
Achromatic  eye-pieces,  1065.  Rainbows,  primary,  secondary,  and  supernumerary, 
1066 pp.  1031-1086. 

Sundry  additions;  goniometers,  convex  lenses,  nodal  points,  .     .     .     .pp.  1086-1088*. 

CHAPTER  LXXIII.    COLOUR. 

Colour  of  opaque  bodies,  1067.  Of  transparent  bodies.  Superposition  of  coloured  glasses, 
1068.  Colours  of  mixed  powders,  1069.  Mixture  of  coloured  lights.  Different 
compositions  may  produce  the  same  visual  impression,  1 070.  Methods  of  mixture : 


viii  TABLE   OF   CONTENTS. 

glass  plate;  rotating  disc ;  overlapping  spectra.  A  mixtui-e  may  be  either  a  mean  or 
a  sum.  Colour  equations,  1071.  Helmholtz's  crossed  sUts,  and  Maxwell's  colour- 
box,  1072.  Results  of  observation :  substitution  of  similars ;  personal  differences ; 
any  four  colours  are  connected  by  one  equation;  any  five  colours  yield  one  match  by 
taking  means.  Sum  of  colours  analogous  to  resultant  of  forces.  Mean  analogous  to 
centre  of  gravity,  1073.  Cone  of  colour.  Hue,  depth,  and  brightness.  Complemen- 
taries.  All  hues  except  purple  are  spectral,  1074.  Three  primary  colour-sensations, 
red,  green,  and  violet,  1075.  Accidental  images,  negative  and  positive,  1076. 
Colour-blind  vision  is  dichroic,  the  red  primary  being  wanting,  1077.  Colour  and 
musical  pitch,  1078 pp.  1087-1098. 

CHAPTER  LXXIV.    WAVE  THEORY  OF  LIGHT. 

Principle  of  Huygens.  Wave-front,  1079.  Explanation  of  rectilinear  propagation. 
Spherical  wave-surface  in  isotropic  medium.  Two  wave-surfaces  in  non-isotropic 
medium,  1080.  Construction  for  wave-front  in  refraction.  Law  of  sines,  1081. 
Reflection,  1082.  Newtonian  explanation  of  refraction.  Foucault's  crucial  experi- 
ment, 1083.  Principle  of  least  time.  Application  to  reflection  and  refraction. 
More  exact  statement  of  the  principle.  Application  to  foci  and  caustics.  2^s  a 
minimum  or  maximum,  1083.  Application  to  terrestrial  refraction.  Rays  in  air  are 
concave  towards  the  denser  side,  1084.  Calculation  of  curvature  of  horizontal  or 
nearly  horizontal  rays,  1085.  Of  inclined  rays,  1086.  Astronomical  refraction,  1088. 
Mirage,  1089.  Curved  rays  of  sound,  1090.  Calculation  of  their  curvature,  1091. 
Diffraction  fringes,  1092.  Gratings,  1093.  Principle  of  diffraction  spectrum,  1094. 
Practical  application,  and  deduction  of  wave-lengths,  1095.  Retardation  gratings, 
1096.  Reflection  gratings,  1097.  Standard  spectrum,  1098.  Wave-lengths,  1099. 
Colours  of  thin  films,  1100, pp.  1099-1118. 


CHAPTEB  LXXV.    POLARIZATION  AND  DOUELE  REFRACTION. 

Experiment  of  two  tourmalines.  Polarizer  and  analyser,  1101.  Polarization  by  reflec- 
tion. The  transmitted  light  also  polarized.  Polarizing  angle,  1102.  .Plane  of 
polarization,  1103.  Polarization  by  double  refraction,  1104.  Explanation  of  double 
refraction  in  uniaxal  crystals.  Wave-surface  for  ordinary  ray  spherical,  for  extra- 
ordinary ray  spheroidal.  Extraordinary  index.  Property  of  tourmaline,  1105. 
Nicol's  prism,  1106.  Colours  produced  by  thin  plates  of  selenite,  1107.  Rectilinear 
vibration  changed  to  elliptic.  Analogy  of  Lissajous'  figures.  Resolution  of  elliptic 
vibration  by  analyser.  Why  the  light  is  coloured.  Why  a  thick  plate  shows  no 
colour.  Crossed  plates,  1108.  Plate  perpendicular  to  axis  shows  rings  and  cross, 
1109.  Explanation,  1110.  Crystals  are  isotropic,  uniaxal,  or  biaxal,  1111.  Rota- 
tion of  plane  of  polarization.  Quartz  and  sugar.  Production  of  colour,  1112. 
Magneto-optic  rotation.  Kerr's  results,  1113.  Circular  polarization  a  case  of  ellip- 
tic. Quarter- wave  plates.  Fresnel's  rhomb.  How  to  distinguish  circularly-polarized 
from  common  light,  1114.  Discussion  as  to  direction  of  vibration  in  plane-polarized 
light,  1115.  Vibrations  of  ordinary  light,  1116.  Polarization  of  dark  heat-rays, 
1117, pp.  1119-1137. 

EXAMPLES   IN  ACOUSTICS,          pp.  1138-1140. 

EXAMPLES  IN  OPTICS, pp  1140-1144. 

ANSWERS  TO  EXAMPLES, pp.  1144-1145. 


FRENCH   AND  ENGLISH  MEASURES. 


A   DECIMETRE   DIVIDED   INTO  CENTIMETRES   AND   MILLIMETRES. 


,  '    i  2— i  ;l    i  *  ml  1  nli  'I.   'I ,  'i    ,   ; 

1 1 1 1  n  1 1  i  n  n  1 1 1  i  i  1 1 1 1 1 , 1 1 1 1  1 1 1 1  1 1 1 1  1 1  1 1 1  1 1  n  1 1  i  1 1 1 1  i  i  1 1  1 1  1 1 1 1 1  i  i  1 1 1  1 1 1  M  1 1 1 1  1 1  1 1 1  1 1 1  1 1 1  I 


INCHES    AND   TENTE 


REDUCTION  OF  FRENCH  TO  ENGLISH  MEASURES. 


LENGTH. 

1  millimetre  =  '03937  inch,  or  about  ^\  inch. 
1  centimetre  =  '3937  inch. 
1  decimetre =3 -937  inch. 
1  metre=39'37  inch=3'231  ft.  =  l'0936  yd. 
1  kilometre=1093-6  yds.,  or  about  f  mile. 
More  accurately,  1  metre =39 '370432  in. 
=3-2808693  ft.=l '09362311  yd. 

AREA. 

Isq.millim.  =-00155  sq.  in.. 
1  sq.  centim.  =  -155  sq.  in. 
1  sq.  decim.  =15 '5  sq.  in. 
sq.  metre  =  1550  sq.  in.  =  10764  sq.  ft.  = 
1-196  sq.  yd. 

VOLUME. 

1  cub.  millim.  =  -000061  cub.  in. 

1  cub.  centim.  =  '061025  cub.  in. 

1  cub.  decim.  =61 '0254  cub.  in. 

cub.  metre=61025  cub.  in.=35'3156  cub. 

ft. =1'308  cub.  yd. 


The  Litre  (used  for  liquids)  is  the  same  as 
the  cubic  decimetre,  and  is  equal  to  17617 
pint,  or  -22021  gallon. 

MASS  AND  WEIGHT. 

1  milligramme=  -01543  grain. 

1  gramme         =15'432  grain. 

1  kilogramme  =  15432  grains =2 -205  Ibs.  avoir. 

More  accurately,  the  kilogramme  is 

2-20462125  Ibs. 

MISCELLANEOUS. 

1  gramme  per  sq.  centim.      =2'0481  Ibs.  p«r 

sq.  ft. 
1  kilogramme  per  sq.  centim.  =14 -223  Ibs.  per 

sq.  in. 

1  kilogrammetre  =  7-2331  foot-pounds. 
1  force  de  cheval  =  75  kilogrammetres  per 
second,  or  542$  foot-pounds  per  second  nearly, 
whereas  1   horse-power  (English I =550  foot- 
pounds per  second. 


REDUCTION  TO  C.G.S.   MEASURES.     (See  page  48.) 
[cm.  denotes  centimetre(s);  gin,  denotes  gramme(s).] 


LENGTH. 

1  incn  =2'54  centimetres,  nearly. 

1  foot  =30-48  centimetres,  nearly. 

1  yard  =91 '44  centimetres,  nearly. 

1  statute  mile  =160933  centimetres,  nearly. 
More  accurately,  1  inch=2'5399772  centi- 
metres. 

AREA. 

1  sq.  inch  =6-45  sq.  cm.,  nearly. 
1  sq.  foot  =929  sq.  cm.,  nearly. 
1  sq.  yard =8361  sq.  cm.,  nearly. 
1  sq.  mile  =2'59  x  1010  sq.  cm.,  nearly. 

VOLUME. 

1  cub.  inch  =16-39  cub.  cm.,  nearly. 
1  cub.  foot  =23316  cub.  cm.,  nearly. 


1  cub.  yard =764535  cub.  cm.,  nearly. 
1  gallon       =4541  cub.  cm.,  nearly. 

MASS. 

1  grain  =  -0648  gramme,  nearly. 
1  oz.  avoir.  =  28 '35  gramme,  nearly. 
1  Ib.  avoir.  =  453-6  gramme,  nearly. 
1  ton  =1-016  x  106  gramme,  nearly 

More  accurately,  1  Ib.  avoir,  =453 '59265  gm 

VELOCITY. 

1  mile  per  hour         =447.04  cm.  per  sec. 
1  kilometre  per  Eour=277  cm.  per  sec. 

DENSITY. 
1  Ib.  per  cub.  foot       =  -016019  gm.  per  cub. 

CHI, 

62-4  Ibs.  per  cub.  ft    =1  gm.  per  cub.  cm. 


FRENCH  AND   ENGLISH   MEASURES. 


FORCE  (assuming  ^=981).     (See  p.  48.) 
Weight  of  1  grain      =  63  '57  dynes,  nearly. 
„     loz.  avoir.  =278  x  lOMynes.uearty. 
„     1  Ib.  avoir.  =  4'45  x  lO'dynes.nearly. 
„     1  ton         =  9  '97  x  108  dynes.nearly. 
„     1  gramme  =981  dynes,  nearly. 
„     1  kilogramme  =  9 '81  x  108  dynes, 
nearly. 

WORK  (assuming  ^=981).    (See  p.  48.) 
1  foot-pound  =  l'356x!07  ergs,  nearly. 

1  kilogrammetre      =9 '81  x  107  ergs,  nearly. 
Work  in  a  second  ) 

by  one  theoretical  V  =7'46  x  109  ergs,  nearly, 
"horse." 


STRESS  (assuming  ^r =981). 

1  Ib.  per  sq.  ft.        =479  dynes  per  sq.  cm., 

nearly. 
1  Ib.  per  sq.  inch      =6'9  x  10*  dynes  per  sq. 

cm.,  nearly. 
1  kilog.  per  sq.  cm.  =9'81  x  105  dynes  per  sq. 

cm.,  nearly. 

760  mm.  of  mercury  at  0°C. =1'014  x  106  dynes 
per  sq.  cm.,  nearly. 

30  inches  of  mercury  at  0°  C.  =1 '0163  x  10s 

dynes  per  sq.  cm.,  nearly. 

1  inch  of  mercury  at  0°  C.  =3'388  x  104  dynes 

per  sq.  cm.,  nearly. 


TABLE   OF  CONSTANTS. 

The  velocity  acquired  in  falling  for  one  second  in  vacuo,  in  any  part  of  Great  Britain,  is 
about  32'2  feet  per  second,  or  9 '81  metres  per  second. 

The  pressure  of  one  atmosphere,  or  760  millimetres  (29 '922  inches)  of  mercury,  is  1*033 
kilogramme  per  sq.  centimetre,  or  14'73  Ibs.  per  square  inch. 

The  weight  of  a  litre  of  dry  air,  at  this  pressure  (at  Paris)  and  0°  C.,  is  1-293  gramme. 

The  weight  of  a  cubic  centimetre  of  water  is  about  1  gramme. 

The  weight  of  a  cubic  foot  of  water  is  about  62'  4  Ibs. 

The  equivalent  of  a  unit  of  heat,  in  gravitation  units  of  energy,  is — 

772  for  the  foot  and  Fahrenheit  degree. 
1390  for  the  foot  and  Centigrade  degree. 

424  for  the  metre  and  Centigrade  degree. 
42400  for  the  centimetre  and  Centigrade  degree. 

In  absolute  units  of  energy,  the  equivalent  is — 

41-6  millions  for  the  centimetre  and  Centigrade  degree; 
or  1  gramme-dogree  is  equivalent  to  41  '6  million  ergs. 


ACOUSTICS. 


CHAPTER    LXII. 


PRODUCTION   AND   PROPAGATION   OF   SOUND. 

886.  Sound  is  a  Vibration. — Sound,  as  directly  known  to  us  by 
the  sense  of  hearing,  is  an  impression  of  a  peculiar  character,  very 
broadly  distinguished  from  the  impressions  received  through  the 
rest  of  our  senses,  and  admitting  of  great 
variety  in  its  modifications.  The  at- 
tempt to  explain  the  physiological  ac- 
tions which  constitute  hearing  forms  no 
part  of  our  present  design.  The  business 
of  physics  is  rather  to  treat  of  those 
external  actions  which  constitute  sound, 
considered  as  an  objective  existence  ex- 
ternal to  the  ear  of  the  percipient. 

It  can  be  shown,  by  a  variety  of  ex- 
periments, that  sound  is  the  result  of 
vibratory  movement.  Suppose,  for  ex- 
ample, we  fix  one  end  C  of  a  straight 
spring  C  D  (Fig.  592)  in  a  vice  A,  then 
draw  the  other  end  D  aside  into  the 
position  D',  and  let  it  go.  In  virtue  of 
its  elasticity  (§  126),  the  spring  will  re- 
turn to  its  original  position;  but  the 
kinetic  energy  which  it  acquires  in  re- 
turning is  sufficient  to  carry  it  to  a  nearly  Fjg.  593. -vibration  of  straight  spring, 
equal  distance  on  the  other  side;  and  it 

thus  swings  alternately  from  one  side  to  the  other  through  distances 
very  gradually  diminishing,  until  at  last  it  comes  to  rest.  Such 
movement  is  called  vibratory.  The  motion  from  D'  to  D",  or  from 
D"  to  D',  is  called  a  single  vibration.  The  two  together  constitute  a 

55 


866 


PRODUCTION  AND  PROPAGATION  OF  SOUND. 


double  or  complete  vibration;  and  the  time  of  executing  a  complete 
vibration  is  the  period  of  vibration.  The  amplitude  of  vibration 
for  any  point  in  the  spring  is  the  distance  of  its  middle  position 
from  one  of  its  extreme  positions.  These  terms  have  been  already 
employed  (§  107)  in  connection  with  the  movements  of  pendulums 
to  which  indeed  the  movements  of  vibrating  springs  bear  an  extremely 
close  resemblance.  The  property  of  isochronism,  which  approximately 
characterizes  the  vibrations  of  the  pendulum,  also  belongs  to  the 
spring,  the  approximation  being  usually  so  close  that  the  period  may 
practically  be  regarded  as  altogether  independent  of  the  amplitude. 


Fig.  593.— Vibration  of  BelL 

When  the  spring  is  long,  the  extent  of  its  movements  may  gene- 
rally be  perceived  by  the  eye.  In  consequence  of  the  persistence  of 
impressions,  we  see  the  spring  in  all  its  positions  at  once;  and  the 
edges  of  the  space  moved  over  are  more  conspicuous  than  the  central 
parts,  because  the  motion  of  the  spring  is  slowest  at  its  extreme 
positions. 

As  the  spring  is  lowered  in  the  vice,  so  as  to  shorten  the  vibrating 
portion  of  it,  its  movements  become  more  rapid,  and  at  the  same  time 


VIBRATION   OF  A  PLATE. 


867 


more  limited,  until,  when  it  is  very  short,  the  eye  is  unable  to  detect 
any  sign  of  motion.  But  where  sight  fails  us,  hearing  comes  to  our 
aid.  As  the  vibrating  part  is  shortened  more  and  more,  it  emits  a 
musical  note,  which  continually  rises  in  pitch;  and  this  effect  con- 
tinues after  the  movements  have  become  much  too  small  to  be  visible. 

It  thus  appears  that  a  vibratory  movement,  if  sufficiently  rapid, 
may  produce  a  sound.  The  following  experiments  afford  additional 
illustration  of  this  principle,  and  are  samples  of  the  evidence  from 
which  it  is  inferred  that  vibratory  movement  is  essential  to  the  pro- 
duction of  sound. 

Vibration  of  a  Bell. — A  point  is  fixed  on  a  stand,  in  such  a  posi- 
tion as  to  be  nearly  in  contact  with  a  glass  bell  (Fig.  593).  If  a 
rosined  fiddle-bow  is  then  drawn  over  the  edge  of  the  bell,  until  a 


Fig.  6SM.— Vibration  of  Plate. 

musical  note  is  emitted,  a  series  of  taps  are  heard,  due  to  the  striking 
of -the  bell  against  the  point.  A  pith-ball,  hung  by  a  thread,  is 
driven  out  by  the  bell,  and  kept  in  oscillation  as  long  as  the  sound 
continues.  By  lightly  touching  the  bell,  we  may  feel  that  it  is 
vibrating;  and  if  we  press  strongly,  the  vibration  and  the  sound 
will  both  be  stopped. 

Vibration  of  a  Plate. — Sand  is  strewn  over  the  surface  of  a  hori- 
zontal plate  (Fig.  594),  which  is  then  made  to  vibrate  by  drawing  a 


868 


PRODUCTION  AND   PROPAGATION   OF   SOUND. 


bow  over  its  edge.  As  soon  as  the  plate  begins  to  sound,  the  sand 
dances,  leaves  certain  parts  bare,  and  collects  in  definite  lines,  which 
are  called  nodal  lines.  These  are,  in  fact,  the  lines  which  separate 
portions  of  the  plate  whose  movements  are  in  oppo- 
site directions.  Their  position  changes  whenever  the 
plate  changes  its  note. 

The  vibratory  condition  of  the  plate  is  also  mani- 
fested by  another  phenomenon,  opposite — so  to  speak 
— to  that  just  described.  If  very  fine  powder,  such  as 
lycopodium,  be  mixed  with  the  sand,  it  will  not  move 
with  the  sand  to  the  nodal  lines,  but  will  form  little 
heaps  in  the  centre  of  the  vibrating  segments;  and 
these  heaps  will  be  in  a  state  of  violent  agitation,  with 
more  or  less  of  gyratory  movement,  as  long  as  the 
plate  is  vibrating.  This  phe- 
nomenon, after  long  baffling 
explanation,  was  shown  by 
Faraday  to  be  due  to  indraughts 
of  air,  and  ascending  currents,  vibration  of  string. 
brought  about  by  the  move- 
ments of  the  plate.  In  a  moderately  good 
vacuum,  the  lycopodium  goes  with  the  sand 
to  the  nodal  lines. 

Vibration  of  a  String. — When  a  note  is 
produced  from  a  musical  string  or  wire,  its 
vibrations  are  often  of  sufficient  amplitude 
to  be  detected  by  the  eye.  The  string  thus 
assumes  the  appearance  of  an  elongated  spindle 
(Fig.  595). 

Vibration  of  the  Air. — The  sonorous  body 
may  sometimes  be  air,  as  in  the  case  of  organ- 
pipes,  which  we  shall  describe  in  a  later  chap- 
ter. It  is  easy  to  show  by  experiment  that 
when  a  pipe  speaks,  the  air  within  it  is  vibrat- 
ing. Let  one  side  of  the  tube  be  of  glass, 
and  let  a  small  membrane  m,  stretched  over 
a  frame,  be  strewed  with  sand,  and  lowered 
into  the  pipe.  The  sand  will  be  thrown  into 

violent  agitation,  and  the  rattling  of  the  grains,  as  they  fall  back  on 
the  membrane,  is  loud  enough  to  be  distinctly  heard. 


Fig.  596.— Vibration  of  Air. 


SINGING  FLAMES. 


869 


Singing  Flames. — An  experiment  on  the  production  of  musical 
sound  by  flame,  lias  long  been  known  under  the  name  of  the  chemi- 
cal harmonica.  An  appera- 
tus  for  the  production  of 
hydrogen  gas  (Fig.  597)  is 
furnished  with  a  tube,  which 
tapers  off  nearly  to  a  point  at 
its  upper  end,  where  the  gas 
issues  and  is  lighted.  When 
a  tube,  open  at  both  ends, 
is  held  so  as  to  surround  the 
flame,  a  musical  tone  is 
heard,  which  varies  with  the 
dimensions  of  the  tube,  and 
often  attains  considerable 
power.  The  sound  is  due  to 
the  vibration  of  the  air  and 
products  of  combustion  with- 
in the  tube;  and  on  observ- 
ing the  reflection  of  the 
flame  in  a  mirror  rotating 
about  a  vertical  axis,  it  will 
be  seen  that  the  flame  is 
alternately  rising  and  falling, 

its  successive  images,  as  drawn  out  into  a  horizontal  series  by  the 
rotation  of  the  mirror,  resembling  a  number  of  equidistant  tongues 
of  flame,  with  depressions  between  them.  The  experiment  may  also 
be  performed  with  ordinary  coal-gas. 

Trevelyan  Experiment. — A  fire-shovel  (Fig.  598)  is  heated,  and 
balanced  upon  the  edges  of  two  sheets  of  lead  fixed  in  a  vice;  it  is 
then  seen  to  execute  a  series  of  small  oscillations — each  end  being 
alternately  raised  and  depressed — and  a  sound  is  at  the  same  time 
emitted.  The  oscillations  are  so  small  as  to  be  scarcely  perceptible 
in  themselves;  but  they  can  be  rendered  very  obvious  by  attaching 
to  the  shovel  a  small  silvered  mirror,  on  which  a  beam  of  light  is 
directed.  The  reflected  light  can  be  made  to  form  an  image  upon  a 
screen,  and  this  image  is  seen  to  be  in  a  state  of  oscillation  as  long  as 
the  sound  is  heard. 

The  movements  observed  in  this  experiment  are  due  to  the  sudden 
expansion  of  the  cold  lead.  When  the  hot  iron  comes  in  contact  with 


Fig.  597.— Chemical  H« 


870  PRODUCTION  AND  PROPAGATION   OF  SOUND. 

it,  a  protuberance  is  instantly  formed  by  dilatation,  and  the  iron  is 
thrown  up.    It  then  comes  in  contact  with  another  portion  of  the 


Fig.  698. — Trevelyan  Experiment. 

lead,  where  the  same  phenomenon  is  repeated  while  the  first  point 
cools.  By  alternate  contacts  and  repulsions  at  the  two  points,  the 
shovel  is  kept  in  a  continual  state  of  oscillation,  and  the  regular 
succession  of  taps  produces  the  sound. 

The  experiment  is  more  usually  performed  with  a  special  instru- 
ment invented  by  Trevelyan,  and  called  a  rocker,  which,  after  being 
heated  and  laid  upon  a  block  of  lead,  rocks  rapidly  from  side  to  side, 
and  yields  a  loud  note. 

867.  Distinctive  Character  of  Musical  Sound. — It  is  not  easy  to 
draw  a  sharp  line  of  demarcation  between  musical  sound  and  mere 
noise.  The  name  of  noise  is  usually  given  to  any  sound  which  seems 
unsuited  to  the  requirements  of  music. 

This  unfitness  may  arise  from  one  or  the  other  of  two  causes. 
Either, 

1.  The  sound  may  be  unpleasant  from  containing  discordant  ele- 
ments which  jar  with  one  another,  as  when  several  consecutive  keys 
on  a  piano  are  put  down  together.     Or, 

2.  It  may  consist  of  a  confused  succession  of  sounds,  the  changes 
being  so  rapid  that  the  ear  is  unable  to  identify  any  particular  note. 
This  kind  of  noise  may  be  illustrated  by  sliding  the  finger  along  a 
violin-string,  while  the  bow  is  applied. 

All  sounds  may  be  resolved  into  combinations  of  elementary  musi- 
cal tones  occurring  simultaneously  and  in  succession.  Hence  the 
study  of  musical  sounds  must  necessarily  form  the  basis  of  acoustics. 

Every  sound  which  is  recognized  as  musical  is  characterized  by 
what  may  be  called  smoothness,  evenness,  or  regularity;  and  the 
physical  cause  of  this  regularity  is  to  be  found  in  the  accurate 


VEHICLE   OF   SOUND 


871 


periodicity  of  the  vibratory  movements  which  produce  the  sound. 
By  periodicity  we  mean  the  recurrence  of  precisely  similar  states  at 
equal  intervals  of  time,  so  that  the  movements  exactly  repeat  them- 
selves; and  the  time  which  elapses  between  two  successive  recur- 
rences of  the  same  state  is  called  the  period  of  the  movements. 

Practically,  musical  and  unmusical  sounds  often  shade  insensibly 
into  one  another.  The  tones  of  every  musical  instrument  are  accom- 
panied by  more  or  less  of  unmusical  noise.  The  sounds  of  bells  and 
drums  have  a  sort  of  intermediate  character;  and  the  confused  as- 
semblage of  sounds  which  is  heard  in  the  streets  of  a  city  blends  at 
a  distance  into  an  agreeable  hum. 

868.  Vehicle  of  Sound.  —  The  origin  of  sound  is  always  to  be  found 
in  the  vibratory  movements  of  a  sonorous  body;  but  these  vibratory 
movements  cannot  bring  about  the  sensation  of  hearing  unless  there 
be  a  medium  to  transmit  them  to  the  auditory  apparatus.  This 
medium  may  be  either  solid,  liquid,  or  gaseous,  but  it  is  necessary 
that  it  be  elastic.  A  body  vibrating  in  an  absolute  vacuum,  or  in  a 
medium  utterly  destitute  of  elasticity, 
would  fail  to  excite  our  sensations  of 
hearing.  This  assertion  is  justified  by 
the  following  experiments:  — 

1.  Under  the  receiver  of  an  air-pump 
is  placed  a  clock-work  arrangement  for 
producing  a  number  of  strokes  on  a  bell. 

It  is  placed  on  a  thick  cushion  of  felt, 
or  other  inelastic  material,  and  the  air 
in  the  receiver  is  exhausted  as  com- 
pletely as  possible.  If  the  clock-work 
is  then  started  by  means  of  the  handle 
g,  the  hammer  will  be  seen  to  strike  the 
bell,  but  the  sound  will  be  scarcely  aud- 
ible. If  hydrogen  be  introduced  into 
the  vacuum,  and  the  receiver  be  again 
exhausted,  the  sound  will  be  much  more 
completely  extinguished,  being  heard 
with  difficulty  even  when  the  ear  is 

placed  in  contact  with  the  receiver.  Hence  it  may  fairly  be  con- 
cluded that  if  the  receiver  could  be  perfectly  exhausted,  and  a  per- 
fectly inelastic  support  could  be  found  for  the  bell,  no  sound  at  all 
would  be  emitted. 


599.  -sound  in  Exhausted  Receiver. 


872  PRODUCTION   AND   PROPAGATION   OF   SOUND. 

2.  The  experiment  may  be  varied  by  using  a  glass  globe,  furnished 
with  a  stop-cock,  and  having  a  little  bell  suspended  within  it  by  a 
thread.  If  the  globe  is  exhausted  of  air,  the  sound  of  the  bell  will 
be  scarcely  audible.  The  globe  may  be  filled  with 
any  kind  of  gas,  or  with  vapour  either  saturated  or 
non-saturated,  and  it  will  thus  be  found  that  all 
these  bodies  transmit  sound. 

Sound  is  also  transmitted  through  liquids,  as  may 
easily  be  proved  by  direct  experiment.    Experiment, 
however,  is  scarcely  necessary  for  the  establishment 
of  the  fact,  seeing  that  fishes  are  provided  with  audi- 
Fig.  eoo.          tory  apparatus,  and  have  often  an  acute  sense  of 

Globe  with  Stop-cock.  J  . 

hearing. 

As  to  solids,  some  well-known  facts  prove  that  they  transmit 
sound  very  perfectly.  For  example,  light  taps  with  the  head  of  a 
pin  on  one  end  of  a  wooden  beam,  are  distinctly  heard  by  a  person 
with  his  ear  applied  to  the  other  end,  though  they  cannot  be  heard 
at  the  same  distance  through  the  air.  This  property  is  sometimes 
•employed  as  a  test  of  the  soundness  of  a  beam,  for  the  experiment 
will  not  succeed  if  the  intervening  wood  is  rotten,  rotten  wood 
being  very  inelastic. 

The  stethoscope  is  an  example  of  the  transmission  of  sound  through 
.solids.  It  is  a  cylinder  of  wood,  with  an  enlargement  at  each  end, 
and  a  perforation  in  its  axis.  One  end  is  pressed  against  the  chest 
of  the  patient,  while  the  observer  applies  his  ear  to  the  other.  He 
is  thus  enabled  to  hear  the  sounds  produced  by  various  internal 
actions,  such  as  the  beating  of  the  heart  and  the  passage  of  the  air 
through  the  tubes  of  the  lungs.  Even  simple  auscultation,  in  which 
jfche  ear  is  applied  directly  to  the  surface  of  the  body,  implies  the 
.transmission  of  sound  through  the  walls  of  the  chest. 

By  applying  the  ear  to  the  ground,  remote  sounds  can  often  be  much 
more  distinctly  heard;  and  it  is  stated  that  savages  can  in  this  way 
obtain  much  information  respecting  approaching  bodies  of  enemies. 

We  are  entitled  then  to  assert  that  sound,  as  it  affects  our  organs 
of  hearing,  is  an  effect  which  is  propagated,  from  a  vibrating  body, 
through  an  elastic  and  ponderable  medium. 

869.  Mode  of  Propagation  of  Sound. — We  will  now  endeavour  to 
explain  the  action  by  which  sound  is  propagated. 

Let  .there  be  a  plate  a  vibrating  opposite  the  end  of  a  long  tube, 
.and  let  us  consider  what  happens  during  the  passage  of  the  plate 


MODE   OF  PROPAGATION   OF  SOUND.  873 

from  its  most  backward  position  a",  to  its  most  advanced  position  a'. 
This  movement  of  the  plate  may  be  divided  in  imagination  into  a 
number  of  successive  parts,  each  of  which  is  communicated  to  the 
layer  of  air  close  in  front  of  it,  which  is  thus  compressed,  and,  in  its 


a."a.o 

I) 


Fig.  601.— Propagation  of  Sound. 


endeavour  to  recover  from  this  compression,  reacts  upon  the  next 
layer,  which  is  thus  in  its  turn  compressed.  The  compression  is  thus 
passed  on  from  layer  to  layer  through  the  whole  tube,  much  in  the 
same  way  as,  when  a  number  of  ivory  balls  are  laid  in  a  row,  if  the 
first  receives  an  impulse  which  drives  it  against  the  second,  each  ball 
will  strike  against  its  successor  and  be  brought  to  rest. 

The  compression  is  thus  passed  on  from  layer  to  layer  through  the 
tube,  and  is  succeeded  by  a  rarefaction  corresponding  to  the  back- 
ward movement  of  the  plate  from  a'  to  a".  As  the  plate  goes  on 
vibrating,  these  compressions  and  rarefactions  continue  to  be  propa- 
gated through  the  tube  in  alternate  succession.  The  greatest  com- 
pression in  the  layer  immediately  in  front  of  the  plate,  occurs  when 
the  plate  is  at  its  middle  position  in  its  forward  movement,  and  the 
greatest  rarefaction  occurs  when  it  is  in  the  same  position  in  its 
backward  movement.  These  are  also  the  instants  at  which  the  plate 
is  moving  most  rapidly.1  When  the  plate  is  in  its  most  advanced 
position,  the  layer  of  air  next  to  it,  A  (Fig.  602)  will  be  in  its  natu- 
ral state,  and  another  layer  at  A,, 
half  a  wave-length  further  on,  will  »'  D.f 
also  be  in  its  natural  state,  the  pulse 
having  travelled  from  A  to  A1}  while 
the  plate  was  moving  from  a"  to  a.  &' 
At  intervening  points  between  A 

and  Aj,  the  layers  will  have  various  amounts  of  compression  corre- 
sponding to  the  different  positions  of  the  plate  in  its  forward  move- 
ment. The  greatest  compression  is  at  C,  a  quarter  of  a  wave-length 
in  advance  of  A,  having  travelled  over  this  distance  while  the  plate 
was  advancing  from  a  to  a.  The  compressions  at  D  and  Dl  repre- 

1  See  §  870,  also  Note  A  at  the  end  of  this  chapter. 


874  PRODUCTION   AND   PROPAGATION   OF   SOUND. 

sent  those  which  existed  immediately  in  front  of  the  plate  when  it 
had  advanced  respectively  one-fourth  and  three-fourths  of  the  dis- 
tance from  a"  to  a,  and  the  curve  A C'  Aa  is  the  graphical  represen- 
tation both  of  condensation  and  velocity  for  all  points  in  the  air 
between  A  and  Ar 

If  the  plate  ceased  vibrating,  the  condition  of  things  now  existing 
in  the  portion  of  air  AAt  would  be  transferred  to  successive  portions 
of  air  in  the  tube,  and  the  curve  AC'A:  would,  as  it  were,  slide 
onward  through  the  tube  with  the  velocity  of  sound,  which  is  about 
1100  feet  per  second.  But  the  plate,  instead  of  remaining  perman- 
ently at  a',  executes  a  backward  movement,  and  produces  rarefactions 
and  retrograde  velocities,  which  are  propagated  onwards  in  the  same 
manner  as  the  condensations  and  forward  velocities.  A  complete 
wave  of  the  undulation  is  accordingly  represented  by  the  curve 
'A,  (Fig.  603),  the  portions  of  the  curve  "below  the  line  of 


Fig.  603.— Graphical  Representation  of  Complete  Wave. 

abscissas  being  intended  to  represent  rarefactions  and  retrograde  velo- 
cities. If  we  suppose  the  vibrating  plate  to  be  rigidly  connected 
with  a  piston  which  works  air-tight  in  the  tube,  the  velocities  of  the 
particles  of  air  in  the  different  points  of  a  wave-length  will  be  iden- 
tical with  the  velocities  of  the  piston  at  the  different  parts  of  its 
motion. 

The  wave-length  A  A2  is  the  distance  that  the  pulse  has  travelled 
while  the  vibrating  plate  was  moving  from  its  most  backward  to  its 
most  advanced  position,  and  back  again.  During  this  time,  which 
is  called  the  period  of  the  vibrations,  each  particle  of  air  goes  through 
its  complete  cycle  of  changes,  both  as  regards  motion  and  density. 
The  period  of  vibration  of  any  particle  is  thus  identical  with  that  of 
the  vibrating  plate,  and  is  the  same  as  the  time  occupied  by  the 
waves  in  travelling  a  wave-length.  Thus,  if  the  plate  be  one  leg  of 
a  common  A  tuning-fork,  making  435  complete  vibrations  per  second, 
the  period  will  be  ^ th  of  a  second,  and  the  undulation  will  travel  in 
this  time  a  distance  of  VW  feet,  or  2  feet  6  inches,  which  is  there- 
fore the  wave-length  in  air  for  this  note.  If  the  plate  continues  to 
vibrate  in  a  uniform  manner,  there  will  be  a  continual  series  of  equal 


NATURE   OF  UNDULATIONS.  875 

and  similar  waves  running  along  the  tube  with  the  velocity  of  sound. 
Such  a  succession  of  waves  constitutes  an  undulation.  Each  wave 
consists  of  a  condensed  portion,  and  a  rarefied  portion,  which  are 
distinguished  from  each  other  in  Fig.  601  by  different  tints,  the 
dark  shading  being  intended  to  represent  condensation. 

870.  Nature  of  Undulations.  —  The  possibility  of  condensations 
and  rarefactions  being  propagated  continually  in  one  direction,  while 
each  particle  of  air  simply  moves  backwards  and  forwards  about  its 
original  position,  is  illustrated  by  Fig.  604,  which  represents,  in  an 


A  B 

C 

D 

E 

F  , 

v 

AB 

C 

D 

E 

F 

Os 

AB 

C 

D 

E 

F 

a/ 

A 

BC 

D 

E 

F 

(t 

A 

BC 

D 

E 

F 

Or 

A 

B 

CD 

E 

F 

a/ 

A 

B 

CD 

E 

F 

a 

'/ 

A 

B 

C 

DE 

F 

a/ 

A 

B 

C 

D 

E  F 

a/ 

A 

B 

C 

D 

EF    ^ 

A 

B 

C 

D 

EF 

Os 

A 

B 

C 

D 

E 

Fa 

A  B 

C 

D 

E 

F  t 

v 

Fig.  604.—  longitudinal  Vibration. 

exaggerated  form,  the  successive  phases  of  an  undulation  propagated 
through  7  particles  ABCDEFa  originally  equidistant,  the  dis- 
tance from  the  first  to  the  last  being  one  wave-length  of  the  undula- 
tion. The  diagram  is  composed  of  thirteen  horizontal  rows,  the  first 
and  last  being  precisely  alike.  The  successive  rows  represent  the 
positions  of  the  particles  at  successive  times,  the  interval  of  time 
from  each  row  to  the  next  being  -^th  of  the  period  of  bhe  un- 
dulation. 

In  the  first  row  A  and  a  are  centres  of  condensation,  and  D  is  a 
centre  of  rarefaction.  In  the  third  row  B  is  a  centre  of  condensa- 
tion, and  E  a  centre  of  rarefaction.  In  the  fifth  row  the  con- 
densation and  rarefaction  have  advanced  by  one  more  letter,  and 
so  on  through  the  whole  series,  the  initial  state  of  things  being 


876        PRODUCTION  AND  PROPAGATION  OF  SOUND. 

reproduced  when  each  of  these  centres  has  advanced  through  a 
wave-length,  so  that  the  thirteenth  row  is  merely  a  repetition  of 
the  first. 

The  velocities  of  the  particles  can  be  estimated  by  the  comparison 
of  successive  rows.  It  is  thus  seen  that  the  greatest  forward  velocity 
is  at  the  centres  of  condensation,  and  the  greatest  backward  velocity 
at  the  centres  of  rarefaction.  Each  particle  has  its  greatest  veloci- 
ties, and  greatest  condensation  and  rarefaction,  in  passing  through 
its  mean  position,  and  comes  for  an  instant  to  rest  in  its  positions  of 
greatest  displacement,  which  are  also  positions  of  mean  density. 

The  distance  between  A  and  a  remains  invariable,  being  always 
a  wave-length,  and  these  two  particles  always  agree  in  phase. 
Any  other  two  particles  represented  in  the  diagram  are  always  in 
different  phases,  and  the  phases  of  A  and  D,  or  B  and  E,  or  C  and  F, 
are  always  opposite;  for  example,  when  A  is  moving  forwards  with 
the  maximum  velocity,  D  is  moving  backwards  with  the  same 
velocity. 

The  vibrations  of  the  particles,  in  an  undulation  of  this  kind,  are 
called  longitudinal;  and  it  is  by  such  vibrations  that  sound  is  pro- 
pagated through  air.  Fig.  605  illustrates  the  manner  in  which  an 
undulation  may  be  propagated  by  means  of  transverse  vibrations, 
that  is  to  say,  by  vibrations  executed  in  a  direction  perpendicular  to 
that  in  which  the  undulation  advances.  Thirteen  particles  A  B  C  D 
EFGHIJKLa  are  represented  in  the  positions  which  they  occupy 
at  successive  times,  whose  interval  is  one-sixth  of  a  period.  At  the 
instant  first  considered,  D  and  J  are  the  particles  which  are  furthest 
displaced.  At  the  end  of  the  first  interval,  the  wave  has  advanced 
two  letters,  so  that  F  and  L  are  now  the  furthest  displaced.  At  the 
end  of  the  next  interval,  the  wave  has  advanced  two  letters  further, 
and  so  on,  the  state  of  things  at  the  end  of  the  six  intervals,  or  of 
one  complete  period,  being  the  same  as  at  the  beginning,  so  that 
the  seventh  line  is  merely  a  repetition  of  the  first.  Some  examples 
of  this  kind  of  wave-motion  will  be  mentioned  in  later  chapters. 

871.  Propagation  in  an  Open  Space. — When  a  sonorous  disturb- 
ance occurs  in  the  midst  of  an  open  body  of  air,  the  undulations  to 
which  it  gives  rise  run  out  in  all  directions  from  the  source.  If  the 
disturbance  is  symmetrical  about  a  centre,  the  waves  will  be  spheri- 
cal; but  this  case  is  exceptional.  A  disturbance  usually  produces 
condensation  on  one  side,  at  the  same  instant  that  it  produces  rare- 
faction on  another.  This  is  the  case,  for  example,  with  a  vibrating 


PROPAGATION   IN   AN   OPEN   SPACE.  877 

plate,  since,  when  it  is  moving  towards  one  side,  it  is  moving  away 
from  the  other.  These  inequalities  which  exist  in  the  neighbour- 
hood of  the  sonorous  body,  have,  however,  a  tendency  to  become  less 
marked,  and  ultimately  to  disappear,  as  the  distance  is  increased. 
Fig.  606  represents  a  diametral  section  of  a  series  of  spherical  waves. 
Their  mode  of  propagation  has  some  analogy  to  that  of  the  circular 

B    c    D   E 

A  G  a, 

H      I     J      K      L 

c    D    E        G    H 

a      c  I      i 

A      B  J      K      L     ^ 

F      G      H       |      , 

E  K      i 

L     ^ 
I     J       K      , 

L     a, 


A 

B 

C 

v 

A 

G 

B 

C 

D 

E 

F 

A 

B 

C 

D 

E 

F 

G 

A 

B 

C 

D 

E 

F 

G 

A 

B 

C 

D 

E 

F 

G 

I     J      K 

Fig.  605.— Transverse  Vibration. 

waves  produced  on  water  by  dropping  a  stone  into  it;  but  the  par- 
ticles which  form  the  waves  of  water  rise  and  fall,  whereas  those 
which  form  sonorous  waves  merely  advance  and  retreat,  their  lines 
of  motion  being  always  coincident  with  the  directions  along  which 
the  sound  travels.  In  both  cases  it  is  important  to  remark  that  the 
undulation  does  not  involve  a  movement  of  transference.  Thus, 
when  the  surface  of  a  liquid  is  traversed  by  waves,  bodies  floating 
on  it  rise  and  fall,  but  are  not  carried  onward.  This  property  is 
characteristic  of  undulations  generally.  An  undulation  may  be 
defined  as  a  system  of  movements  in  which  the  several  particles  move 
to  and  fro,  or  round  and  round,  about  definite  points,  in  such  a 


878         PRODUCTION  AND  PROPAGATION  OF  SOUND. 

manner  as  to  produce  the  continued  onward  transmission  of  a  con- 
dition, or  series  of  conditions. 

There  is  one  important  difference  between  the  propagation  of 
sound  in  a  uniform  tube  and  in  an  open  space.  In  the  former  case, 
the  layers  of  air  corresponding  to  successive  wave-lengths  are  of  equal 


Fig.  606.— Propagation  in  Open  Space. 

mass,  and  their  movements  are  precisely  alike,  except  in  so  far  as 
they  are  interfered  with  by  friction.  Hence  sound  is  transmitted 
through  tubes  to  great  distances  with  but  little  loss  of  intensity, 
especially  if  the  tubes  are  large.1 

The  same  principle  is  illustrated  by  the  ease  with  which  a  scratch 

1  Regnault,  in  his  experiments  on  the  velocity  of  sound,  found  that  in  a  conduit  '108  of 
a  metre  in  diameter,  the  report  of  a  pistol  charged  with  a  gramme  of  powder  ceased  to 
be  heard  at  the  distance  of  1150  metres.  In  a  conduit  of  '3™,  the  distance  was  3810™. 
In  the  great  conduit  of  the  St.  Michel  sewer,  of  lm>10,  the  sound  was  made  by  successive 
reflections  to  traverse  a  distance  of  10,000  metres  without  becoming  inaudible.— D. 


DISSIPATION   OF  SONOROUS  .ENERGY.  879 

on  a  log  of  wood  is  heard  at  the  far  end,  the  substance  of  the  log 
acting  like  the  body  of  air  within  a  tube. 

In  an  open  space,  each  successive  layer  has  to  impart  its  own 
condition  to  a  larger  layer;  hence  there  is  a  continual  diminution  of 
amplitude  in  the  vibrations  as  the  distance  from  the  source  increases. 
This  involves  a  continual  decrease  of  loudness.  An  undulation 
involves  the  onward  transference  of  energy;  and  the  amount  of 
energy  which  traverses,  in  unit  time,  any  closed  surface  described 
about  the  source,  must  be  equal  to  the  energy  which  the  source 
emits  in  unit  time.  Hence,  by  the  reasoning  which  we  employed 
in  the  case  of  radiant  heat  (§  465),  it  follows  that  the  intensity 
of  sonorous  energy  diminishes  according  to  the  law  of  inverse 
squares. 

The  energy  of  a  particle  executing  simple  vibrations  in  obedience 
to  forces  of  elasticity,  varies  as  the  square  of  the  amplitude  of  its 
excursions;  for,  if  the  amplitude  be  doubled,  the  distance  worked 
through,  and  the  mean  working  force,  are  both  doubled,  and  thus 
the  work  which  the  elastic  forces  do  during  the  movement  from 
either  extreme  position  to  the  centre  is  quadrupled.  This  work  is 
equal  to  the  energy  of  the  particle  in  any  part  of  its  course.  At  the 
extreme  positions  it  is  all  in  the  shape  of  potential  energy;  in  the 
middle  position  it  is  all  in  the  shape  of  kinetic  energy;  and  at 
intermediate  points  it  is  partly  in  one  of  these  forms,  and  partly  in 
the  other. 

It  can  be  shown  that  exactly  half  the  energy  of  a  complete  wave 
is  kinetic,  the  other  half  being  potential. 

872.  Dissipation  of  Sonorous  Energy. — The  reasoning  by  which  we 
have  endeavoured  to  establish  the  law  of  inverse  squares,  assumes 
that  onward  propagation  involves  no  loss  of  sonorous  energy.  This 
assumption  is  not  rigorously  true,  inasmuch  as  vibration  implies 
friction,  and  friction  implies  the  generation  of  heat,  at  the  expense 
of  the  energy  which  produces  the  vibrations.  Sonorous  energy  must 
therefore  diminish  with  distance  somewhat  more  rapidly  than  accord- 
ing to  the  law  of  inverse  squares.  All  sound,  in  becoming  extinct, 
becomes  converted  into  heat. 

This  conversion  is  greatly  promoted  by  defect  of  homogeneity 
in  the  medium  of  propagation.  In  a  fog,  or  a  snow-storm,  the 
liquid  or  solid  particles  present  in  the  air  produce  innumerable 
reflections,  in  each  of  which  a  little  sonorous  energy  is  converted 
into  heat. 


880  PRODUCTION   AND   PROPAGATION   OF   SOUND. 

873.  Velocity  of  Sound  in  Air. — The  propagation  of  sound  through 
an  elastic  medium  is  not  instantaneous,  but  occupies  a  very  sensible 
time  in  traversing  a  moderate  distance.  For  example,  the  flash  of  a 
gun  at  the  distance  of  a  few  hundred  yards  is  seen  some  time  before 
the  report  is  heard.  The  interval  between  the  two  impressions 
may  be  regarded  as  representing  the  time  required  for  the  propa- 
gation of  the  sound  across  the  intervening  distance,  for  the  time 
occupied  by  the  propagation  of  light  across  so  small  a  distance  is 
inappreciable. 

It  is  by  experiments  of  this  kind  that  the  velocity  of  sound  in  air 
has  been  most  accurately  determined.  Among  the  best  determina- 
tions may  be  mentioned  that  of  Lacaille,  and  other  members  of  a 
commission  appointed  by  the  French  Academy  in  1738;  that  of 
Arago,  Bouvard,  and  other  members  of  the  Bureau  de  Longitudes 
in  1822;  and  that  of  Moll,  Vanbeek,  and  Kuytenbrouwer  in  Holland, 
in  the  same  year.  All  these  determinations  were  obtained  by  firing 
cannon  at  two  stations,  several  miles  distant  from  each  other,  and 
noting,  at  each  station,  the  interval  between  seeing  the  flash  and 
hearing  the  sound  of  the  guns  fired  at  the  other.  If  guns  were  fired 
only  at  one  station,  the  determination  would  be  vitiated  by  the  effect 
of  wind  blowing  either  with  or  against  the  sound.  The  error  from 
this  cause  is  nearly  eliminated  by  firing  the  guns  alternately  at  the 
two  stations,  and  still  more  completely  by  firing  them  simultaneously 
This  last  plan  was  adopted  by  the  Dutch  observers,  the  distance  of 
the  two  stations  in  their  case  being  about  nine  miles.  Regnault  has 
quite  recently  repeated  the  investigation,  taking  advantage  of  the 
important  aid  afforded  by  modern  electrical  methods  for  registering 
the  times  of  observed  phenomena.  All  the  most  careful  determina- 
tions agree  very  closely  among  themselves,  and  show  that  the  velo- 
city of  sound  through  air  at  0°  C.  is  about  332  metres,  or  1090  feet 
per  second.1  The  velocity  increases  with  the  temperature,  being 
proportional  to  the  square  root  of  the  absolute  temperature  by  air 
thermometer  (§  325).  If  t  denote  the  ordinary  Centigrade  tempera- 

1  A  recent  determination  by  Mr.  Stone  at  the  Cape  of  Good  Hope  is  worthy  of  note  as 
being  based  on  the  comparison  of  observations  made  through  the  sense  of  hearing  alone. 
It  had  previously  been  suggested  that  the  two  senses  of  sight  and  hearing,  which  are  con- 
cerned in  observing  the  flash  and  report  of  a  cannon,  might  not  be  equally  prompt  in  re- 
ceiving impressions  (Airy  on  Sound,  p.  131).  Mr.  Stone  accordingly  placed  two  observers 
— one  near  a  cannon,  and  the  other  at  about  three  miles  distance ;  each  of  whom  on  hear- 
ing the  report,  gave  a  signal  through  an  electric  telegraph.  The  result  obtained  was  in 
precise  agreement  with  that  stated  in  the  text. 


VELOCITY   OF   SOUND   IN   AIR.  881 

ture,  and  o  the  coefficient  of  expansion  -003G6,  the  velocity  of  sound 
through  air  at  any  temperature  is  given  by  the  formula 


332  \/l  +  a  t  in  metres  per  second,  or 
1090  VI  +  a  <  iQ  feet  per  second. 

The  actual  velocity  of  sound  from  place  to  place  on  the  earth's  sur- 
face is  found  by  compounding  this  velocity  with  the  velocity  of  the 
wind. 

There  is  some  reason,  both  from  theory  and  experiment,  for  be- 
lieving that  very  loud  sounds  travel  rather  faster  than  sounds  of 
moderate  intensity. 

874.  Theoretical  Computation  of  Velocity.  —  By  applying  the  prin- 
ciples of  dynamics  to  the  propagation  of  undulations,1  it  is  computed 
that  the  velocity  of  sound  through  air  must  be  given  by  the  formula 


D  denoting  the  density  of  the  air,  and  E  its  coefficient  of  elasticity, 
as  measured  by  the  quotient  of  pressure  applied  by  compression 
produced. 

Let  P  denote  the  pressure  of  the  air  in  units  of  force  per  unit  of 
area;  then,  if  the  temperature  be  kept  constant  during  compression, 
a  small  additional  pressure  p  will,  by  Boyle's  law,  produce  a  com- 

pression equal  to  p,  and  the  value  of  E,  being  the  quotient  of  p  by 

this  quantity,  will  be  simply  P. 

On  the  other  hand,  if  no  heat  is  allowed  either  to  enter  or  escape, 
the  temperature  of  the  air  will  be  raised  by  compression,  and  addi- 
tional resistance  will  thus  be  encountered.  In  this  case,  as  shown 
in  §  500,  the  coefficient  of  elasticity  will  be  P  k,  k  denoting  the 
ratio  of  the  two  specific  heats,  which  for  air  and  simple  gases  is 
about  1-41. 

It  thus  appears  that  the  velocity  of  sound  in  air  cannot  be  less  than 

y/£  nor  greater  than  ^/1'41  ?.    Its  actual  velocity,  as  determined 

by  -observation,  is  identical,  or  practically  identical,  with  the  latter 
of  these  limiting  values.  Hence  we  must  infer  that  the  compressions 
and  extensions  which  the  particles  of  air  undergo  in  transmitting 
sound  are  of  too  brief  duration  to  allow  of  any  sensible  transference 
of  heat  from  particle  to  particle. 

This  conclusion  is  confirmed  by  another  argument  due  to  Professor 

1  See  note  B  at  the  end  of  this  chapter. 
50 


882        PRODUCTION  AND  PROPAGATION  OF  SOUND. 

Stokes.  If  the  inequalities  of  temperature  due  to  compression  and 
expansion  were  to  any  sensible  degree  smoothed  down  by  conduction 
and  radiation,  this  smoothing  down  would  diminish  the  amount  of 
energy  available  for  wave-propagation,  and  would  lead  to  a  falling 
off  in  intensity  incomparably  more  rapid  than  that  due  to  the  law 
of  inverse  squares. 

875.  Numerical  Calculation.  —  The  following  is  the  actual  process  of 
calculation  for  perfectly  dry  air  at  0°  C.,  the  centimetre,  gramme,  and 
second  being  taken  as  the  units  of  length,  mass,  and  time. 

The  density  of  dry  air  at  0°,  under  the  pressure  of  1033  grammes 
per  square  centimetre,  at  Paris,  is  '001293  of  a  gramme  per  cubic 
centimetre.  But  the  gravitating  force  of  a  gramme  at  Paris  is  981 
dynes  (§  91).  The  density  '001293  therefore  corresponds  to  a 
pressure  of  1033  x  981  dynes  per  sq.  cm.;  and  the  expression  for  the 
velocity  in  centimetres  per  second  is 


that  is,  332'4  metres  per  second,  or  1093  feet  per  second. 

876.  Effects  of  Pressure,  Temperature,  and  Moisture.  —  The  velocity 
of  sound  is  independent  of  the  height  of  the  barometer,  since  changes 
of  this  element  (at  constant  temperature)  affect  P  and  D  in  the  same 
direction,  and  to  the  same  extent. 

For  a  given  density,  if  P0  denote  the  pressure  at  0°,  and  o  the  co- 
efficient of  expansion  of  air,  the  pressure  at  t°  Centigrade  is  P0  (1  -f-o  t\ 

the  value  of  o  being  about  ^3. 

Hence,  if  the  velocity  at  0°  be  1090  feet  per  second,  the  velocity 
at  t°  will  be  1090  ^/1-h  2-^.  At  the  temperature  50°  F.  or  10°  C., 
which  is  approximately  the  mean  annual  temperature  of  this  country, 
the  value  of  this  expression  is  about  1110,  and  at  86°  F.  or  30°  C.  it 
is  about  1148.  The  increase  of  velocity  is  thus  about  a  foot  per 
second  for  each  degree  Fahrenheit. 

The  humidity  of  air  has  some  influence  on  the  velocity  of  sound, 
inasmuch  as  aqueous  vapour  is  lighter  than  air;  but  the  effect  is 
comparatively  trifling,  at  least  in  temperate  climates.  At  the  tem- 
perature 50°  F.,  air  saturated  with  moisture  is  less  dense  than  dry 
air  by  about  1  part  in  220,  and  the  consequent  increase  of  velocity 
cannot  be  greater  than  about  1  part  in  440,  which  will  be  between 
2  and  3  feet  per  second.  The  increase  should,  in  fact,  be  somewhat 


VELOCITY   IN   GASES  AND   LIQUIDS.  883 

less  than  this,  inasmuch  as  the  value  of  k  (the  ratio  of  the  two  specific 
heats)  appears  to  be  only  1-31  for  aqueous  vapour.1 

877.  Newton's  Theory,  and  Laplace's  Modification.  —  The  earliest 
theoretical  investigation  of  the  velocity  of  sound  was  that  given  by 
Newton  in  the  Principia  (book  2,  section  8).  It  proceeds  on  the 
tacit  assumption  that  no  changes  of  temperature  are  produced  by 
the  compressions  and  extensions  which  enter  into  the  constitution 
of  a  sonorous  undulation;  and  the  result  obtained  by  Newton  is 
equivalent  to  the  formula 


•p 

or  since  (§  21Cf)  ^=gH,  where  H  denotes  the  height  of  a  homo- 
geneous atmosphere,  and  the  velocity  acquired  in  falling  through 
any  height  s  is  V  2  #  s,  the  velocity  of  sound  in  air  is,  according  to 
Newton,  the  same  as  the  velocity  which  ivould  be  acquired  Ij  falling 
in  vacuo  through  half  the  height  of  a  homogeneous  atmosphere. 
This,  in  fact,  is  the  form  in  which  Newton  states  his  result.2 

Newton  himself  was  quite  aware  that  the  value  thus  computed 
theoretically  was  too  small,  and  he  throws  out  a  conjecture  as  to  the 
cause  of  the  discrepancy;  but  the  true  cause  was  first  pointed  out 
by  Laplace,  as  depending  upon  increase  of  temperature  produced  by 
compression,  and  decrease  of  temperature  produced  by  expansion. 

878.  Velocity  in  Gases   generally.  —  The  same   principles  which 
apply  to  air  apply  to  gases  generally;  and  since  for  all  simple  gases 
the  ratio  of  the  two  specific  heats  is  1*41,  the  velocity  of  sound  in 

any  simple  gas  is  A/  1'41  j-,  D  denoting  its  absolute  density  at  the 

pressure  P.  Comparing  two  gases  at  the  same  pressure,  we  see  that 
the  velocities  of  sound  in  them  will  be  inversely  as  the  square  roots 
of  their  absolute  densities;  and  this  will  be  true  whether  the  tem- 
peratures of  the  two  gases  are  the  same  or  different. 

879.  Velocity  of  Sound  in  Liquids.  —  The  velocity  of   sound   in 
water  was  measured  by  Colladon,  in  1826,  at  the  Lake  of  Geneva. 
Two  boats  were  moored  at  a  distance  of  13,500  metres  (between  8 
and  9  miles).     One  of  them  carried  a  bell,  weighing  about  140  Ibs., 
immersed  in  the  lake.     Its  hammer  was  moved  by  an  external  lever, 
so  arranged  as  to  ignite  a  small  quantity  of  gunpowder  at  the  instant 

1  Rankine  on  the  Steam  Engine,  p.  320. 

8  Newton's  investigation  relates  only  to  simple  waves  ;  but  if  these  have  all  the  same 
velocity  (as  Newton  shows),  this  must  also  be  the  velocity  of  the  complex  wave  which  thuy 
compose.  Hence  the  restriction  is  only  apparent. 


884  PRODUCTION   AND   PROPAGATION   OF   SOUND. 

of  striking  the  bell.  An  observer  in  the  other  boat  was  enabled  to 
hear  the  sound  by  applying  his  ear  to  the  extremity  of  a  trumpet- 
shaped  tube  (Fig.  607),  having  its  lower  end  covered  with  a  mem- 
brane and  facing  towards  the  direction  from 
which  the  sound  proceeded.  By  noting  the  in- 
terval between  seeing  the  flash  and  hearing  the 
sound,  the  velocity  with  which  the  sound  tra- 
velled through  the  water  was  determined.  The 
velocity  thus  computed  was  1435  metres  per 
second,  and  the  temperature  of  the  water  was 
8°1  C. 
l-i  COT!  Formula  (1)  of  §  874  holds  for  liquids  as  well 

as  for  gases. 

The  resistance  of  water  to  compression  is  about  21  X 1010  dynes 
per  sq.  cm.,  and  the  correcting  factor  for  the  heat  of  compression,  as 
calculated  by  §  505,  is  T0012,  which  may  be  taken  as  unity.  The 
density  is  also  unity.  Hence  we  have 

v-  VE  =  V(2'!  x  1010)  =  144914  cm.  per  sec., 

that  is,  about  1449  metres  per  second;  which  agrees  sufficiently  well 
with  the  experimental  determination. 

Wertheim  has  measured  the  velocity  of  sound  in  some  liquids  by 
an  indirect  method,  which  will  be  explained  in  a  later  chapter.  He 
finds  it  to  be  1160  metres  per  second  in  ether  and  alcohol,  and  1900 
in  a  solution  of  chloride  of  calcium. 

880.  Velocity  of  Sound  in  Solids. — The  velocity  of  sound  in  cast- 
iron  was  determined  by  Biot  and  Martin  by  means  of  a  connected 
series  of  water-pipes,  forming  a  conduit  of  a  total  length  of  951 
metres.  One  end  of  the  conduit  was  struck  with  a  hammer,  and  an 
observer  at  the  other  end  heard  two  sounds,  the  first  transmitted  by 
the  metal,  and  the  second  by  the  air,  the  interval  between  them 
being  2'5  seconds.  Now  the  time  required  for  travelling  this  dis- 
tance through  air,  at  the  temperature  of  the  experiment  (11°  C.),  is 
2 '8  seconds.  The  time  of  transmission  through  the  metal  was  there- 
fore '3  of  a  second,  which  is  at  the  rate  of  3170  metres  per  second. 
It  is,  however,  to  be  remarked,  that  the  transmitting  body  was  not 
a  continuous  mass  of  iron,  but  a  series  of  376  pipes,  connected  to- 
gether by  collars  of  lead  and  tarred  clotn,  which  must  have  consid- 
erably delayed  the  transmission  of  the  sound.  But  in  spite  of  this, 
the  velocity  is  about  nine  times  as  great  as  in  air. 


VELOCITY   OF   SOUND   IN   SOLIDS. 


885 


Werfcheim,  by  the  indirect  methods  above  alluded  to,  measured 
the  velocity  of  sound  in  a  number  of  solids,  with  the  following  results, 
the  velocity  in  air  being  taken  as  the  unit  of  velocity: — 


Lead 3'974  to  4-120 

Tin, 7-338  to  7'480 

Gold, 5-603  to  6-424 

Silver, 7 '903  to  8'057 

Zinc, 9-863  to  11'009 

Copper, 11-167 

Platinum,  ,     .     .     .     .  7'823  to  8'467 


Steel, 14-361  to  15108 

Iron, 15-108 

Brass, 10-224 

Glass, 14-956  to  16759 

Flint  Glass,  ....  11'890  to  12'220 

Oak, 9-902  to  12-02 

Fir 12-49    to  17'26 


881.  Theoretical  Computation.  —  The  formula  ^/g  serves  for  solids 

as  well  as  for  liquids  and  gases;  but  as  solids  can  be  subjected  to 
many  different  kinds  of  strain,  whereas  liquids  and  gases  can  be 
subjected  to  only  one,  we  may  have  different  values  of  E,  and  dif- 
ferent velocities  of  transmission  of  pulses  for  the  same  solid.  This  is 
true  even  in  the  case  of  a  solid  whose  properties  are  alike  in  all 
directions  (called  an  isotropic  solid)  ;  but  the  great  majority  of  solids 
are  very  far  from  fulfilling  this  condition,  and  transmit  sound  more 
rapidly  in  some  directions  than  in  others. 

When  the  sound  is  propagated  by  alternate  compressions  and 
extensions  running  along  a  substance  which  is  not  prevented  from 
extending  and  contracting  laterally,  the  elasticity  E  becomes  iden- 
tical1 with  Young's  modulus  (§  128).  On  the  other  hand,  if  uniform 
spherical  waves  of  alternate  compression  and  extension  spread  out- 
wards, symmetrically,  from  a  point  in  the  centre  of  an  infinite  solid, 
lateral  extension  and  contraction  will  be  prevented  by  the  symmetry 
of  the  action.  The  effective  elasticity  is,  in  this  case,  greater  than 
Young's  modulus,  and  the  velocity  of  sound  will  be  increased  accord- 
ingly. 

By  the  table  on  p.  79  the  value  of  Young's  modulus  for  copper  is 
120  x  1010,  and  by  the  table  on  p.  xii.  the  density  of  copper  is  about 
8-8.  Hence,  for  the  velocity  of  sound  through  a  copper  rod,  in  centi- 
metres per  second,  we  have 


or  3693  metres  per  second. 

This  is  about  11  '1  times  the  velocity  in  air. 


1  Subject  to  a  very  small  correction  for  heat  of  compression,  which  can  be  calculated  by 
the  formula  of  §  505.     In  the  case  of  iron,  the  correcting  factor  is  about  1'0023. 


PRODUCTION  AND   PROPAGATION   OF  SOUND. 


882.  Reflection  of  Sound. — When  sonorous  waves  meet  a  fixed 
obstacle  they  are  reflected,  and  the  two  sets  of  waves — one  direct, 
and  the  other  reflected — are  propagated  just  as  if  they  came  from 
two  separate  sources.  If  the  reflecting  surface  is  plane,  waves  di- 


Fig.  60S.—  Reflection  of  Sound. 

verging  from  any  centre  0  (Fig.  608)  in  front  of  it  are  reflected  so 
as  to  diverge  from  a  centre  O'  symmetrically  situated  behind  it,  and 
an  ear  at  any  point  M  in  front  hears  the  reflected  sound  as  if  it  came 
from  O'. 

The  direction  from  which  a  sound  appears  to  the  hearer  to  proceed 
is  determined  by  the  direction  along  which  the  sonorous  pulses  are 
propagated,  and  is  always  normal  to  the  waves.  A  normal  to  a  set 
of  sound-waves  may  therefore  conveniently  be  called  a  ray  of  sound. 

O I  is  a  direct  ray,  and  I  M  the 
corresponding  reflected  ray;  and  it 
is  obvious,  from  the  symmetrical 
position  of  the  points  O  0',  that 
these  two  rays  are  equally  in- 
clined to  the  surface,  or  the  angles 
of  incidence  and  reflection  are 
equal. 

883.  Illustrations  of  Reflection  of  Sound. — The  reflection  of  sonorous 
waves  explains  some  well-known  phenomena.  If  aba  (Fig.  609) 
be  an  elliptic  dome  or  arch,  a  sound  emitted  from  either  of  the  foci 
//  will  be  reflected  from  the  elliptic  surface  in  such  a  direction  as  to 
pass  through  the  other  focus.  A  sound  emitted  from  either  focus 


Fig.  609.— Reflection  from  Elliptic  Roof. 


REFLECTION   OF  SOUND.  887 

may  thus  be  distinctly  heard  at  the  other,  even  when  quite  inaudible 
at  nearer  points.  This  is  a  consequence  of  the  property,  that  lines 
drawn  to  any  point  on  an  ellipse  from  the  two  foci  are  equally 
inclined  to  the  curve. 

The  experiment  of  the  conjugate  mirrors  (§  468)  is  also  applicable 
to  sound.   Let  a  watch  be  hung  in  the  focus  of  one  of  them  (Fig.  G10), 


Fig.  610. — Reflection  of  Sound  from  Conjugate  Mirrors. 

and  let  a  person  hold  his  ear  at  the  focus  of  the  other;  or  still  better, 
to  avoid  intercepting  the  sound  before  it  falls  on  the  second  mirror, 
let  him  employ  an  ear-trumpet,  holding  its  further  end  at  the  focus. 
He  will  distinctly  hear  the  ticking,  even  when  the  mirrors  are  many 
yards  apart.1 

884.  Echo. — Echo  is  the  most  familiar  instance  of  the  reflection  of 
sound.  In  order  to  hear  the  echo  of  one's  own  voice,  there  must  be 
a  distant  body  capable  of  reflecting  sound  directly  back,  and  the 
number  of  syllables  that  an  echo  will  repeat  is  proportional  to  the 

1  Sondhaus  has  shown  that  sound,  like  light,  is  capable  of  being  refracted.  A  spherical 
balloon  of  collodion,  filled  with  carbonic  acid  gas,  acts  as  a  sound-lens.  If  a  watch  be 
hung  at  some  distance  from  it  on  one  side,  an  ear  held  at  the  conjugate  focus  on  the  other 
side  will  hear  the  ticking.  See  also  a  later  section  on  "  Curved  Kays  of  Sound  "  in  the 
chapter  on  the  "  Wave  Theory  of  Light." 


888        PRODUCTION  AND  PROPAGATION  OF  SOUND. 

distance  of  this  obstacle.  The  sounds  reflected  to  the  speaker  have 
travelled  first  over  the  distance  0  A  (Fig.  611)  from  him  to  the 
reflecting  body,  and  then  back  from  A  to  O.  Supposing  five  syllables 
to  be  pronounced  in  a  second,  and  taking  the  velocity  of  sound  as 
1100  feet  per  second,  a  distance  of  550  feet  from  the  speaker  to 
the  reflecting  body  would  enable  the  speaker  to  complete  the  fifth 
syllable  before  the  return  of  the  first;  this  is  at  the  rate  of  110  feet 


Fig.  611.— Echo. 

per  syllable.  At  distances  less  than  about  100  feet  there  is  not  time 
for  the  distinct  reflection  of  a  single  syllable;  but  the  reflected 
sound  mingles  with  the  voice  of  the  speaker.  This  is  particularly 
observable  under  vaulted  roofs. 

Multiple  echoes  are  not  uncommon.  They  are  due,  in  some 
cases,  to  independent  reflections  from  obstacles  at  different  dis- 
tances; in  others,  to  reflections  of  reflections.  A  position  exactly 
midway  between  two  parallel  walls,  at  a  sufficient  distance  apart,  is 
favourable  for  the  observance  of  this  latter  phenomenon.  One  of 
the  most  frequently  cited  instances  of  multiple  echoes  is  that  of  the 
old  palace  of  Simonetta,  near  Milan,  which  forms  three  sides  of  a 
quadrangle.  According  to  Kircher,  it  repeats  forty  times. 

885.  Speaking  and  Hearing  Trumpets. — The  complete  explanation 
of  the  action  of  these  instruments  presents  considerable  difficulty. 
The  speaking-trumpet  (Fig.  612)  consists  of  a  long  tube  (sometimes 
6  feet  long),  slightly  tapering  towards  the  speaker,  furnished  at  this 
end  with  a  hollow  mouth-piece,  which  nearly  fits  the  lips,  and  at 


SPEAKING  AND   HEARING  TRUMPETS. 


the  other  with  a  funnel-shaped  enlargement,  called  the  bell,  opening 
out  to  a  width  of  about  a  foot.  It  is  much  used  at 
sea,  and  is  found  very  effectual  in  making  the  voice 
heard  at  a  distance.  The  explanation  usually  given  of 
its  action  is,  that  the  slightly  conical  form  of  the  long 
tube  produces  a  series  of  reflections  in  directions  more 
and  more  nearly  parallel  to  the  axis;  but  this  explana- 
tion fails  to  account  for  the  utility  of  the  bell,  which 
experience  has  shown  to  be  considerable.  It  appears 
from  a  theoretical  investigation  by  Lord  Rayleigh 
that  the  speaking-trumpet  causes  a  greater  total  quan- 
tity of  sonorous  energy  to  be  produced  from  the  same 
expenditure  of  breath.1 

Ear -trumpets  have  various  forms,  as  represented 
in  Fig.  613;  having  little  in  common,  except  that  the 
external  opening  or  bell  is  much  larger  than  the  end 
which  is  introduced  into  the  ear.  Membranes  of  gold- 
beaters' skin  are  sometimes  stretched  across  their 
interior,  in  the  positions  indicated  by  the  dotted  lines 
in  Nos.  4  and  5.  No.  6  consists  simply  of  a  bell  with 
such  a  membrane  stretched  across  its  outer  end,  while  Fig.  en 

Speaking-trumpet. 

its  inner  end  communicates  with  the  ear  by  an  indian- 
rubber  tube  with  an  ivory  end-piece.  These  light  membranes  are 
peculiarly  susceptible  of  impression  from  aerial  vibrations.  In  Reg- 
nault's  experiments  above 
cited,  it  was  found  that  mem- 
branes were  affected  at  dis- 
tances greater  than  those 
at  which  sound  was  heard. 
886.  Interference  of  Sonor- 
ous Undulations. — When 
two  systems  of  waves  are 
traversing  the  same  mat- 
ter, the  actual  motion  of 
each  particle  of  the  matter 
is  the  resultant  of  the  mo- 
tions due  to  each  system 
separately.  When  these 
component  motions  are  in  the  same  direction  the  resultant  is  their 

1  Theory  of  Sound,  vol.  ii.  p.  102. 


Fig.  613.  —Kir-trumpets. 


890  PRODUCTION   AND   PROPAGATION   OF  SOUND. 

sum;  when  they  are  in  opposite  directions  it  is  their  difference;  and 
if  they  are  equal,  as  well  as  opposite,  it  is  zero.  Very  remarkable 
phenomena  are  thus  produced  when  the  two  undulations  have  the 
same,  or  nearly  the  same  wave-length;  and  the  action  which  occurs 
in  this  case  is  called  interference. 

When  two  sonorous  undulations  of  exactly  equal  wave-length 
and  amplitude  are  traversing  the  same  matter  in  the  same  direction, 
their  phases  must  either  be  the  same,  or  must  everywhere  differ  by 
the  same  amount.  If  they  are  the  same,  the  amplitude  of  vibration 
for  each  particle  will  be  double  of  that  due  to  either  undulation 
separately.  If  they  are  opposite — in  other  words,  if  one  undulation 
be  half  a  wave-length  in  advance  of  the  other — the  motions  which 
they  would  separately  produce  in  any  particle  are  equal  and  oppo- 
site, and  the  particle  will  accordingly  remain  at  rest.  Two  sounds 
will  thus,  by  their  conjoint  action,  produce  silence. 

In  order  that  the  extinction  of  sound  may  be  complete,  the  rare- 
fied portions  of  each  set  of  waves  must  be  the  exact  counterparts  of 
the  condensed  portions  of  the  other  set,  a  condition  which  can  only 
be  approximately  attained  in  practice. 

The  following  experiment,  due  to  M.  Desains,  affords  a  very  direct 
illustration  of  the  principle  of  interference.  The  bottom  of  a  wooden 
box  is  pierced  with  an  opening,  in  which  a  powerful  whistle  fits. 
The  top  of  the  box  has  two  larger  openings  symmetrically  placed 
with  respect  to  the  lower  one.  The  inside  of  the  box  is  lined  with 
felt,  to  prevent  the  vibrations  from  being  communicated  to  the  box, 
and  to  weaken  internal  reflection.  When  the  whistle  is  sounded,  if 
a  membrane,  with  sand  strewn  on  it,  is  held  in  various  positions  in 
the  vertical  plane  which  bisects,  at  right  angles,  the  line  joining  the 
two  openings,  the  sand  will  be  agitated,  and  will  arrange  itself  in 
nodal  lines.  But  if  it  is  carried  out  of  this  plane,  positions  will  be 
found,  at  equal  distances  on  both  sides  of  it,  at  which  the  agitation 
is  scarcely  perceptible.  If,  when  the  membrane  is  in  one  of  these 
positions,  we  close  one  of  the  two  openings,  the  sand  is  again  agitated, 
clearly  showing  that  the  previous  absence  of  agitation  was  due  to  the 
interference  of  the  undulations  proceeding  from  the  two  orifices. 

In  this  experiment  the  proof  is  presented  to  the  eye.  In  the  fol- 
lowing experiment,  which  is  due  to  M.  Lissajous,  it  is  presented  to 
the  ear.  A  circular  plate,  supported  like  the  plate  in  Fig.  594,  is 
made  to  vibrate  in  sectors  separated  by  radial  nodes.  The  number 
of  sectors  will  always  be  even,  and  adjacent  sectors  will  vibrate 


INTERFERENCE   OF   SOUND-WAVES.  891 

in  opposite  directions.  Let  a  disk  of  card -board  of  the  same 
size  be  divided  into  the  same  number  of  sectors,  and  let  alternate 
sectors  be  cut  away,  leaving  only  enough  near  the  centre  to  hold  the 
remaining  sectors  together.  If  the  card  be  now  held  just  over  the 
vibrating  disk,  in  such  a  manner  that  the  sectors  of  the  one  are 
exactly  over  those  of  the  other,  a  great  increase  of  loudness  will  be 
observed,  consequent  on  the  suppression  of  the  sound  from  alternate 
sectors;  but  if  the  card-board  disk  be  turned  through  the  width  of 
half  a  sector,  the  effect  no  longer  occurs.  If  the  card  is  made  to 
rotate  rapidly  in  a  continuous  manner,  the  alterations  of  loudness 
will  form  a  series  of  beats. 

It  is  for  a  similar  reason  that,  when  a  large  bell  is  vibrating,  a 
person  in  its  centre  hears  the  sound  as  only  moderately  loud,  while 
within  a  short  distance  of  some  portions  of  the  edge  the  loudness  is 
intolerable. 

887.  Interference  of  Direct  and  Reflected  Waves.1  Nodes  and  Anti- 
nodes. — Interference  may  also  occur  between  undulations  travelling  in 
opposite  directions;  for  example,  between  a  direct  and  a  reflected  sys- 
tem. When  waves  proceeding  along  a  tube  meet  a  rigid  obstacle,  form- 
ing a  cross  section  of  the  tube,  they  are  reflected  directly  back  again, 
the  motion  of  any  particle  close  to  the  obstacle  being  compounded  of 
that  due  to  the  direct  wave,  and  an  equal  and  opposite  motion  due 
to  the  reflected  wave.  The  reflected  waves  are  in  fact  the  images 
(with  reference  to  the  obstacle  regarded  as  a  plane  mirror)  of  the 
waves  which  would  exist  in  the  prolongation  of  the  tube  if  the 
obstacle  were  withdrawn.  At  the  distance  of  half  a  wave-length 
from  the  obstacle  the  motions  due  to  the  direct  and  reflected  waves 
will  accordingly  be  equal  and  opposite,  so  that  the  particles  situated 
at  this  distance  will  be  permanently  at  rest;  and  the  same  is  true  at 
the  distance  of  any  number  of  half  wave-lengths  from  the  obstacle. 
The  air  in  the  tube  will  thus  be  divided  into  a  number  of  vibrating 
segments  separated  by  nodal  planes  or  cross  sections  of  no  vibra- 
tion arranged  at  distances  of  half  a  wave-length  apart.  One  of  these 
nodes  is  at  the  obstacle  itself.  At  the  centres  of  the  vibrating  seg- 
ments— that  is  to  say,  at  the  distance  of  a  quarter  wave-length  plus 
any  number  of  half  wave-lengths  from  the  obstacle  or  from  any  node 
— the  velocities  due  to  the  direct  and  reflected  waves  will  be  equal 
and  in  the  same  direction,  and  the  amplitude  of  vibration  will  ac- 
cordingly be  double  of  that  due  to  the  direct  wave  alone.  These 

1  See  note  C,  page  895. 


892  PRODUCTION   AND   PROPAGATION  OF  SOUND. 

are  the  sections  of  greatest  disturbance  as  regards  change  of  place. 
We  shall  call  them  antinodes.  On  the  other  hand,  it  is  to  be  remem- 
bered that  motion  with  the  direct  wave  is  motion  against  the  re- 
flected waves,  and  vice  versa,  so  that  (§  869)  at  points  where  the 
velocities 'due  to  both  have  the  same  absolute  direction  they  corre- 
spond to  condensation  in  the  case  of  one  of  these  undulations,  and 
to  rarefaction  in  the  case  of  the  other.  Accordingly,  these  sections  of 
maximum  movement  are  the  places  of  no  change  of  density;  and  on 
the  other  hand,  the  nodes  are  the  places  where  the  changes  of  density 
are  greatest.  If  the  reflected  undulation  is  feebler  than  the  direct 
one,  as  will  be  the  case,  for  example,  if  the  obstacle  is  only  imper- 
fectly rigid,  the  destruction  of  motion  at  the  nodes  and  of  change  of 
density  at  the  antinodes  will  not  be  complete;  the  former  will  merely 
be  places  of  minimum  motion,  and  the  latter  of  minimum  change  of 
density. 

Direct  experiments  in  verification  of  these  principles,  a  wall  being 
the  reflecting  body,  were  conducted  by  Savart,  and  also  by  Seebeck, 
the  latter  of  whom  employed  a  testing  apparatus  called  the  acoustic 
pendulum.  It  consists  essentially  of  a  small  membrane  stretched  in 
a  frame,  from  the  top  of  which  hangs  a  very  light  pendulum,  with 
its  bob  resting  against  the  centre  of  the  membrane.  In  the  middle 
portions  of  the  vibrating  segments  the  membrane,  moving  with  the 
air  on  its  two  faces,  throws  back  the  pendulum,  while  it  remains 
nearly  free  from  vibration  at  the  nodes. 

Regnault  made  extensive  use  of  the  acoustic  pendulum  in  his  ex- 
periments on  the  velocity  of  sound.  The  pendulum,  when  thrown 
back  by  the  membrane,  completed  an  electric  circuit,  and  thus 
effected  a  record  of  the  instant  when  the  sound  arrived. 

888.  Beats  Produced  by  Interference. — When  two  notes  which 
are  not  quite  in  unison  are  sounded  together,  a  peculiar  palpitating 
effect  is  produced; — we  hear  a  series  of  bursts  of  sound,  with  inter- 
vals of  comparative  silence  between  them.  The  bursts  of  sound  are 
called  beats,  and  the  notes  are  said  to  beat  together.  If  we  have  the 
power  of  tuning  one  of  the  notes,  we  shall  find  that  as  they  are 
brought  more  nearly  into  unison,  the  beats  become  slower,  and'  that, 
as  the  departure  from  unison  is  increased,  the  beats  become  more 
rapid,  till  they  degenerate  first  into  a  rattle,  and  then  into  a  discord. 
The  effect  is  most  striking  with  deep  notes. 

These  beats  are  completely  explained  by  the  principle  of  interfer- 
ence. As  the  wave-lengths  of  the  two  notes  are  slightly  different, 


BEATS.  893 

while  the  velocity  of  propagation  is  the  same,  the  two  systems  of 
waves  will,  in  some  portions  of  their  course,  agree  in  phase,  and  thus 
strengthen  each  other;  while  in  other  parts  they  will  be  opposite  in 
phase,  and  will  thus  destroy  each  other.  Let  one  of  the  notes,  for 
example,  have  100  vibrations  per  second,  and  the  other  101.  Then, 
if  we  start  from  an  instant  when  the  maxima  of  condensation  from 
the  two  sources  reach  the  ear  together,  the  next  such  conjunction 
will  occur  exactly  a  second  later.  During  the  interval  the  maxima 
of  one  system  have  been  gradually  falling  behind  those  of  the  other, 
till,  at  the  end  of  the  second,  the  loss  has  amounted  to  one  wave- 
length. At  the  middle  of  the  second  it  will  have  amounted  to  half 
a  wave-length,  and  the  two  sounds  will  destroy  each  other.  We 
shall  thus  have  one  beat  and  one  extinction  in  each  second,  as  a 
consequence  of  the  fact  that  the  higher  note  has  made  one  vibration 
more  than  the  lower.  In  general,  the  frequency  of  beats  is  the  dif- 
ference of  the  frequencies  of  vibration  of  the  beating  notes. 


NOTE  A.    §  809. 

That  the  particles  which  are  moving  forward  are  in  a  state  of  compression,  may  be 
shown  in  the  following  way:  —  Consider  an  imaginary  cross  section  travelling  forward 
through  the  tube  with  the  same  velocity  as  the  undulation.  Call  this  velocity  v,  and  the 
velocity  of  any  particle  of  air  «.  Also  let  the  density  of  any  particle  be  denoted  by  p. 
Then  u  and  p  remain  constant  for  the  imaginary  moving  section,  and  the  mass  of  air  which 
it  traverses  in  its  motion  per  unit  time  is  (v  -  u)  p.  As  there  is  no  permanent  transfer  of 
air  in  either  direction  through  the  tube,  the  mass  thus  traversed  must  be  the  same  as  if 
the  air  were  at  rest  at  its  natural  density.  Hence  the  value  of  (a  -  u)  p  is  the  same  for 
all  cross  sections  ;  whence  it  follows,  that  where  u  is  greatest  p  must  be  greatest,  and 
where  u  is  negative  p  is  less  than  the  natural  density. 

If  po  denote  the  natural  density,  we  have  (v  -  u)  p  =  v  p^,  whence  -  =P^J*;  that  is  to  say, 

v         p 

the  ratio  of  the  velocity  of  a  particle  to  the  velocity  of  the  undulation  is  equal  to  the  conden- 
sation existing  at  the  particle.  If  u  is  negative  —  that  is  to  say,  if  the  velocity  be  retrograde 
—  its  ratio  to  v  is  a  measure  of  the  rarefaction. 

From  this  principle  we  may  easily  derive  a  formula  for  the  velocity  of  sound,  bearing  in 
mind  that  u  is  always  very  small  in  comparison  with  v. 

For,  consider  a  thin  lamina  of  air  whose  thickness  is  Sx,  and  let  Su,  Sp,  and  tip  be 
the  excesses  of  the  velocity,  density,  and  pressure  on  the  second  side  of  the  lamina 
above  those  on  the  first  at  the  same  moment.  The  above  equation,  (v-u)  p  =  vpo,  gives 

(v  -  u)  Sp-pSu  =  0,  whence  —  =  v~u,  Or,  since  u  may  be  neglected  in  comparison  with  v, 
Sp        p 


_ 
Sp  ~  p' 

upies  in 
time  the  velocity  of  the  lamina  changes  by  the  amount  -  8  u,  since  the  velocity  on  the 


The  time  which  the  moving  section  occupies  in  traversing  the  lamina  is  —,  and  in  this 


894  PRODUCTION   AND   PROPAGATION   OF  SOUND. 

second  side  of  the  lamina  is  u  +  8u  at  the  beginning  and  w  at  the  end  of  the  time.  The 
force  producing  this  change  of  velocity  (if  the  section  of  the  tube  be  unity)  is  -  5  p,  or 

- 1'41  ?  8p,  and  must  be  equal  to  the  quotient  of  change  of  momentum  by  time,  that  is 
P 

to  -pSx.  Su-^-X-,  or  to  -pvSu.     Hence     "=1'41  -2~.     Equating  this  to  the  othtr 
v  d  p  p  o 

expression  for  —,  we  have 
tp 

?=1-41    P  ,  t»  =  l-41^. 

p  p*v  p 

This  investigation  is  due  to  Professor  Rankine,  Phil.  Trans.  1869. 

NOTE  B.    §  874. 

The  following  is  the  usual  investigation  of  the  velocity  of  transmission  of  sound  through 
a  uniform  tube  filled  with  air,  friction  being  neglected :  Let  x  denote  the  original  distance 
of  a  particle  of  air  from  the  section  of  the  tube  at  which  the  sound  originates,  and  x  +  y  its 
distance  at  time  t,  so  that  y  is  the  displacement  of  the  particle  from  the  position  of  equili- 
brium. Then  a  particle  which  was  originally  at  distance  x  +  Sx  will  at  time  t  be  at  the 
distance  x  +  Sx  +  y  +  8y ;  and  the  thickness  of  the  intervening  lamina,  which  was  originally 

Sx,ia  now  Sx  +  Sy.    Its  compression  is  therefore  - ^  or  ultimately  - 5?,  and  if  P  denote 

ox  ax 

the  original  pressure,  the  increase  of  pressure  is  -1'41  P  --V.     The  excess  of  pressure 

ax 

behind  a  lamina  Sx  above  the  pressure  in  front  is  —  (1'41  P  d^)Sx,  or  1-41  P  -^  Sx; 

ax  ^  ax  dy? 

and  if  D  denote  the  original  density  of  the  air,  the  acceleration  of  the  lamina  will  be  the 
quotient  of  this  expression  by  D .  Sx.  But  this  acceleration  is  -— J?.  Hence  we  have  the 
equation 

d(*~         D  dx3' 
the  integral  of  which  is 

y  =  F  (x 

where  v  denotes    */ 1-41  ?,  and  F,  /denote  any  functions  whatever. 

The  term  F  (x-v  t)  represents  a  wave,  of  the  form  y  =  ~F  (x),  travelling  forwards  with 
velocity  v ;  for  it  has  the  same  value  for  ^  +  S  t  and  xl  +  v .  S  t  as  for  ^  and  x..  The  term 
f(x  +  vt)  represents  a  wave,  of  the  form  y=f  (x),  travelling  backwards  with  the  same 
velocity. 

In  order  to  adapt  this  investigation,  as  well  as  that  given  in  Note  A,  to  the  propagation 
of  longitudinal  vibrations  through  any  elastic  material,  whether  solid,  liquid,  or  gaseous,  we 
have  merely  to  introduce  E  in  the  place  of  1'41  P,  E  denoting  the  coefficient  of  elasticity 

of  the  substance,  as  defined  by  the  condition  that  a  compression  -¥  is  produced  by  a  force 

dx 

(per  unit  area)  of  E  ^. 
dx 

Nora  C.    §  887. 

The  following  is  the  regular  mathematical  investigation  of  the  interference  of  direct 
and  reflected  waves  of  the  simplest  type,  in  a  uniform  tube. 


STATIONARY   UNDULATION.  895 

Using  x,  y,  and  t  in  the  same  sense  as  in  Note  B,  and  measuring  x  from  the  reflecting 
surface  to  meet  the  incident  waves,  we  have,  for  the  incident  waves, 

yi  =  asin^2,,  (1) 

a  dsnoting  the  amplitude,  and  X  the  wave-length.     For  the  reflected  waves,  we  have 

since  this  equation  represents  waves  equal  and  opposite  to  the  former,  and  satisfies  the 
condition  that  at  the  reflecting  surface  (where  x  is  zero)  the  total  disturbance  yi  +  y%  is 
zero.  Putting  y  for  yi  +  yt,  we  have,  by  adding  the  above  equations  and  employing  a  well- 
known  formula  of  trigonometry, 

y  =  2asin  ?2jr  .  cos^27r.  (3) 

A  A 

The  extension  (or  compression  if  negative)  is  -~,  and  we  have 

^=_^2acos*  27r.cos^27r.  (4) 

The  factor  sin  -  2  it  vanishes  at  the  points  for  which  x  is  either  zero  or  a  multiple  of 
A 

^  X,  and  attains  its  greatest  values  (in  arithmetical  sense)  at  those  for  which  x  is  \  X,  or 
\  X  plus  a  multiple  of  £  X.  On  the  other  hand,  the  factor  qos  ^  2  TT  vanishes  at  the  latter 

points,  and  attains  its  greatest  values  at  the  former.     The  points  for  which  sin  —  2  r 

X 

vanishes  are  the  nodes,  since  at  these  points  y  is  constantly  zero;  and  the  points  for 
which  cos  ^-  2  TT  vanishes  are  the  antinodes,  since  at  these  the  extension  or  compression  is 

constantly  zero. 

The  motion  represented  by  equation  (3)  is  the  simplest  type  of  stationary  undulation. 

NOTE  D.     §  888. 

The  following  is  the  mathematical  investigation  of  beats  for  two  systems  of  waves  of 
equal  amplitude  but  slightly  different  wave-length  and  period,  travelling  with  the  same 
velocity. 

o  _  o  _ 

Denote  x-vt  by  9,  —  by  m^  and  -—  by  m*  X,  X2  being  the  wave-lengths  of  the  two 
A!  \.t 

systems,  and  let  their  common  amplitude  be  a.  Then  the  resultant  of  the  two  sets  is 
represented  by 

y  —  a  sin  ml  9  +  a  sin  m*  9 
=  2  a  sin  i  (wii  +  wij)  0 .  cos  J  (OT!  -  mj)  9. 

By  hypothesis  nij  —  r/ij  is  very  small  compared  with  mi  +  m^;  hence  the  factor  cos  J  (mt  —  m2)  9 
remains  nearly  constant  for  an  increment  of  9  which  causes  ^  (mi  +  m?)  9  to  increase  by  2ir. 
The  expression  therefore  represents  a  series  of  waves  having  a  wave-length  intermediate 
between  \  and  X2  (since  ^(mi  +  m^  is  intermediate  between  m^  and  m.2),  and  having  an 
amplitude  2acosi(mi-ntj)0  which  gradually  varies  between  the  limits  zero  and  2  a. 


CHAPTEE    LXIII. 


NUMERICAL   EVALUATION   OF   SOUND. 


889.  Qualities  of  Musical  Sound. — Musical  tones  differ  one  from 
another  in  respect  of  three  qualities; — loudness,  pitch,  and  character. 

Loudness. — The  loudness  of  a  sound  considered  subjectively  is  the 
intensity  of  the  sensation  with  which  it  affects  the  organs  of  hearing. 
Regarded  objectively,  it  depends,  in  the  case  of  sounds  of  the  same 
pitch  and  character,  upon  the  energy  of  the  aerial  vibrations  in  the 
neighbourhood  of  the  ear,  and  is  proportional  to  the  square  of  the 
amplitude. 

Our  auditory  apparatus  is,  however,  so  constructed  as  to  be  more 
susceptible  of  impression  by  sounds  of  high  than  of  low  pitch.  A 
bass  note  must  have  much  greater  energy  of  vibration  than  a  treble 
note,  in  order  to  strike  the  ear  as  equally  loud.  The  intensity  of 
sonorous  vibration  at  a  point  in  the  air  is  therefore  not  an  absolute 
measure  of  the  intensity  of  the  sensation  which  will  be  received  by 
an  ear  placed  at  the  point. 

The  word  loud  is  also  frequently  applied  to  a  source  of  sound,  as 
when  we  say  a  loud  voice,  the  reference  being  to  the  loudness  as 
heard  at  a  given  distance  from  the  source.  The  diminution  of  loud- 
ness  with  increase  of  distance  according  to  the  law  of  inverse  squares 
is  essentially  connected  with  the  proportionality  of  loudness  to  square 
of  amplitude. 

Pitch. — Pitch  is  the  quality  in  respect  of  which  an  acute  sound 
differs  from  a  grave  one;  for  example,  a  treble  note  from  a  bass  note. 
All  persons  are  capable  of  appreciating  differences  of  pitch  to  some 
extent,  and  the  power  of  forming  accurate  judgments  of  pitch  con- 
stitutes what  is  called  a  musical  ear. 

Physically,  pitch  depends  solely  on  frequency  of  vibration,  that 
is  to  say,  on  the  number  of  vibrations  executed  per  unit  time.  In 


QUALITIES   OF   SOUNDS.  897 

ordinary  circumstances  this  frequency  is  the  same  for  the  source  of 
sound,  the  medium  of  transmission,  and  the  drum  of  the  ear  of  the 
person  hearing;  and  in  general  the  transmission  of  vibrations  from 
one  body  or  medium  to  another  produces  no  change  in  their  fre- 
quency. The  second  is  universally  employed  as  the  unit  of  time  in 
treating  of  sonorous  vibrations;  so  that  frequency  means  number  of 
vibrations  per  second.  Increase  of  frequency  corresponds  to  eleva- 
tion of  pitch. 

Period  and  frequency  are  reciprocals.  For  example,  if  the  period 
of  each  vibration  is  -j-J-g-  of  a  second,  the  number  of  vibrations  per 
second  is  100.  Period  therefore  is  an  absolute  measure  of  pitch,  and 
the  longer  the  period  the  lower  is  the  pitch. 

The  wave-length  of  a  note  in  any  medium  is  the  distance  which 
sound  travels  in  that  medium  during  the  period  corresponding  to  the 
note.  Hence  wave-length  may  be  taken  as  a  measure  of  pitch,  pro- 
vided the  medium  be  given;  but,  in  passing  from  one  medium  to 
another,  wave-length  varies  directly  as  the  velocity  of  sound.  The 
wave-length  of  a  given  note  in  air  depends  upon  the  temperature  of 
the  air,  and  is  shortened  in  transmission  from  the  heated  air  of  a 
concert-room  to  the  colder  air  outside,  while  the  pitch  undergoes  no 
change. 

If  we  compare  a  series  of  notes  rising  one  above  another  by  what 
musicians  regard  as  equal  differences  of  pitch,  their  frequencies  will 
not  be  equidifferent,  but  will  form  an  increasing  geometrical  pro- 
gression, and  their  periods  (and  wave-lengths  in  a  given  medium) 
will  form  a  decreasing  geometrical  progression. 

Character. — Musical  sounds  may,  however,  be  alike  as  regards  pitch 
and  loudness,  and  may  yet  be  easily  distinguishable.  We  speak  of 
the  quality  of  a  singer's  voice,  and  the  tone  of  a  musical  instrument; 
and  we  characterize  the  one  or  the  other  as  rich,  sweet,  or  mellow; 
on  the  one  hand:  or  as  poor,  harsh,  nasal,  &c.,  on  the  other.  These 
epithets  are  descriptive  of  what  musicians  call  timbre — a  French 
word  literally  signifying  stamp.  German  writers  on  acoustics  denote 
the  same  quality  by  a  term  signifying  sound-tint.  It  might  equally 
well  be  called  sound-flavour.  We  adopt  character  as  the  best 
English  designation. 

Physically  considered,  as  wave-length  and   wave-amplitude  fall 

under  the  two  previous  heads,  character  must  depend  upon  the  only 

remaining  point   in  which  aerial  waves  can  differ — namely  their 

form,  meaning  by  this  term  the  law  according  to  which  the  velo- 

57 


898  NUMERICAL  EVALUATION   OF  SOUND. 

cities  and  densities  change  from  point  to  point  of  a  wave.  This 
subject  will  be  more  fully  treated  in  Chapter  Ixv.  Every  musical 
sound  is  more  or  less  mingled  with  non-musical  noises,  such  as  puffing, 
scraping,  twanging,  hissing,  rattling,  &c.  These  are  not  compre- 
hended under  timbre  or  character  in  the  usage  of  the  best  writers 
on  acoustics.  The  gradations  of  loudness  which  characterize  the 
commencement,  progress,  and  cessation  of  a  note,  and  upon  which 
musical  effect  often  greatly  depends,  are  likewise  excluded  from  this 
designation.  In  distinguishing  the  sounds  of  different  musical  in- 
struments, we  are  often  guided  as  much  by  these  gradations  and 
extraneous  accompaniments  as  by  the  character  of  the  musical  tones 
themselves. 

890.  Musical  Intervals. — When  two  notes  are  heard,  either  simul- 
taneously or  in  succession,  the  ear  experiences  an  impression  of  a 
special  kind,  involving  a  perception  of  the  relation  existing  between 
them  as  regards  difference  of  pitch.     This  impression  is  often  recog- 
nized as  identical  where  absolute  pitch  is  very  different,  and  we 
express  this  identity  of  impression  by  saying  that  the  musical  inter- 
val is  the  same. 

Each  musical  interval,  thus  recognized  by  the  ear  as  constituting 
a  particular  relation  between  two  notes,  is  found  to  correspond  to  a 
paiticular  ratio  between  their  frequencies  of  vibration.  Thus  the 
octave,  which  of  all  intervals  is  that  which  is  most  easily  recognized 
by  the  ear,  is  the  relation  between  two  notes  whose  frequencies  are 
as  1  to  2,  the  upper  note  making  twice  as  many  vibrations  as  the 
lower  in  any  given  time. 

It  is  the  musician's  business  so  to  combine  sounds  as  to  awaken 
emotions  of  the  peculiar  kind  which  are  associated  with  works  of 
art.  In  attaining  this  end  he  employs  various  resources,  but  musical 
intervals  occupy  the  foremost  place.  It  is  upon  the  judicious  employ- 
ment of  these  that  successful  composition  mainly  depends. 

891.  Gamut. — The  gamut  or  diatonic  scale  is  a  series  of  eight 
notes  having  certain  definite  relations  to  one  another  as  regards 
frequency  of  vibration.     The  first  and  last  of  the  eight  are  at  an 
interval  of  an  octave  from  each  other,  and  are  called  by  the  same 
name;  and  by  taking  in  like  manner  the  octaves  of  the  other  notes 
of  the  series,  we  obtain  a  repetition  of  the  gamut  both  upwards  and 
downwards,  which  may  be  continued  over  as  many  octaves  as  we 
please. 

The  notes  of  the  gamut  are  usually  called  by  the  names 


GAMUT  AND  MUSICAL   INTERVALS.  899 

Do        He        Mi        Fa        Sol        La         Si         Do, 

and  their  vibration-frequencies  are  proportional  to  the  numbers 

19  6  4.3  515  2 

•s         T         t        f        f .       TT 

or,  clearing  fractions,  to 

24         27         30         32          36         40         45  48 

The  intervals  from  Do  to  each  of  the  others  in  order  are  called  a 
second,  a  major  third,  a  fourth,  a  fifth,  a  sixth,  a  seventh,  and  an 
octave  respectively.  The  interval  from  La  to  Do2  is  called  a  minor 
third,  and  is  evidently  represented  by  the  ratio  -f-. 

The  interval  from  Do  to  Re,  from  Fa  to  Sol,  or  from  La  to  Si,  is 
represented  by  the  ratio  f,  and  is  called  a  major  tone.  The  interval 
from  Re  to  Mi,  or  from  Sol  to  La,  is  represented  by  the  ratio  V°>  and 
is  called  a  minor  tone.  The  interval  from  Mi  to  Fa,  or  from  Si  to 
Do2,  is  represented  by  the  ratio  -^f-,  and  is  called  a  limma.  As  the 
square  of  if-  is  a  little  greater  than  f,  a  lirnma  is  rather  more  than 
half  a  major  tone. 

The  intervals  between  the  successive  notes  of  the  gamut  are  ac- 
cordingly represented  by  the  following  ratios1: — 

Do        Re        Mi        Fa        Sol        La        Si        Do* 

9  10  16  9  10  9  16 

F  ~¥~         XT          F  IT          ~5          TJ 

Do  (with  all  its  octaves)  is  called  the  key-note  of  the  piece  of  music, 
and  may  have  any  pitch  whatever.  In  order  to  obtain  perfect  har- 
mony, the  above  ratios  should  be  accurately  maintained  whatever 
the  key-note  may  be. 

892.  Tempered  Gamut. — A  great  variety  of  keys  are  employed  in 
music,  and  it  is  a  practical  impossibility,  at  all  events  in  the  case  of 
instruments  like  the  piano  and  organ,  which  have  only  a  definite  set 
of  notes,  to  maintain  these  ratios  strictly  for  the  whole  range  of  pos- 
sible key-notes.  Compromise  of  some  kind  becomes  necessary,  and 
different  systems  of  compromise  are  called  different  temperaments  or 
different  modes  of  temperament.  The  temperament  which  is  most 
in  favour  in  the  present  day  is  the  simplest  possible,  and  is  called 
equal  temperament,  because  it  favours  no  key  above  another,  but 
makes  the  tempered  gamut  exactly  the  same  for  all.  It  ignores  the 

1  The  logarithmic  differences,  which  are  accurately  proportional  to  the  intervals,  are 
approximately  as  under,  omitting  superfluous  zeros. 

Do        Re        Mi        Fa        Sol        La        Si        Do 
51         46         28         51         46         51         28 


900 


NUMERICAL  EVALUATION   OF  SOUND. 


difference  between  major  and  minor  tones,  and  makes  the  limma 
exactly  half  of  either.  The  interval  from  Do  to  Do2  is  thus  divided 
into  5  tones  and  2  semitones,  a  tone  being  £  of  an  octave,  and  a 
semitone  TV  of  an  octave.  The  ratio  of  frequencies  corresponding  to 
a  tone  will  therefore  be  the  sixth  root  of  2,  and  for  a  semitone  it  will 
be  the  12th  root  of  2. 

The  difference  between  the  natural  and  the  tempered  gamut  for  the 
key  of  C  is  shown  by  the  following  table,  which  gives  the  number 
of  complete  vibrations  per  second  for  each  note  of  the  middle  octave 
of  an  ordinary  piano: — 


Tempered  Gamut.  Natural  Gamut. 
C    .    .      2587  2587 

D    .    .      290-3  291-0 

E    .    .      325-9  323-4 

F    .  345-3  344-9 


Tempered  Gamut.  Natural  Gamut 
G    .     .       387-6  388-0 

A     .     .       435-0  4311 

B     .    .       488-2  485-0 

C     .    .       517-3  517-3 


The  absolute  pitch  here  adopted  is  that  of  the  Paris  Conservatoire, 
and  is  fixed  by  the  rule  that  A  (the  middle  A  of  a  piano,  or  the  A 
string  of  a  violin)  is  to  have  435  complete  vibrations  per  second  in 
the  tempered  gamut.  This  is  rather  lower  than  the  concert-pitch 
which  has  prevailed  in  this  country  in  recent  years,  but  is  probably 
not  so  low  as  that  which  prevailed  in  the  time  of  Handel.  It  will 
be  noted  that  the  number  of  vibrations  corresponding  to  C  is 
approximately  equal  to  a  power  of  2  (256  or  512).  Any  power  of 
2  accordingly  expresses  (to  the  same  degree  of  approximation)  the 
number  of  vibrations  corresponding  to  one  of  the  octaves  of  C. 

The  Stuttgard  congress  (1834)  recommended  528  vibrations  per 
second  for  C,  and  the  C  tuning-forks  sold  under  the  sanction  of  the 
Society  of  Arts  are  guaranteed  to  have  this  pitch.  By  multiplying 
the  numbers  24,  27  ...  48,  in  §  891,  by  11,  we  shall  obtain  the 
frequencies  of  vibration  for  the  natural  gamut  in  C  corresponding  to 
this  standard.  What  is  generally  called  concert-pitch  gives  C  about 
538.  The  C  of  the  Italian  Opera  is  546.  Handel's  C  is  said  to  have 
been  499f 

893.  Limits  of  Pitch  employed  in  Music.— The  deepest  note  re- 
gularly employed  in  music  is  the  C  of  32  vibrations  per  second 
which  is  emitted  by  the  longest  pipe  (the  16-foot  pipe)  of  most 
organs.  Its  wave-length  in  air  at  a  temperature  at  which  the  velo- 
city of  sound  is  1120  feet  per  second,  is  ijgs.=3o  feet.  The  highest 
note  employed  seldom  exceeds  A,  the  third  octave  of  the  A  above 
denned.  Its  number  of  vibrations  per  second  is  435  x  23=3480,  and 


MINOR  AND   PYTHAGOREAN   SCALES.  901 

its  wave-length  in  air  is  about  4  inches.  Above  this  limit  it  is  diffi- 
cult to  appreciate  pitch,  but  notes  of  at  least  ten  times  this  number 
of  vibrations  are  audible. 

The  average  compass  of  the  human  voice  is  about  two  octaves. 
The  deep  F  of  a  bass-singer  has  87,  and  the  upper  G  of  the  treble 
775  vibrations  per  second.  Voices  which  exceed  either  of  these 
limits  are  regarded  as  deep  or  high. 

894.  Minor  Scale  and  Pythagorean  Scale. — The  difference  between 
a  major  and  minor  tone  is  expressed  by  the  ratio  f£,  and  is  called  a 
comma.  The  difference  between  a  minor  tone  and  a  limma  is  ex- 
pressed by  the  ratio  ff ,  and  is  the  smallest  value  that  can  be  assigned 
to  the  somewhat  indefinite  interval  denoted  by  the  name  semitone, 
the  greatest  value  being  the  limma  itself  (y^).  The  signs  #  and  t? 
(sharp  and  flat)  appended  to  a  note  indicate  that  it  is  to  be  raised  or 
lowered  by  a  semitone.  The  major  scale  or  gamut,  as  above  given, 
is  modified  in  the  following  way  to  obtain  the  minor  scale: — 

Do        Re        Mi>         Fa        Sol        La)        Si>        Do* 
I         It  V         f         18-  *          V 

the  numbers  in  the  second  line  being  the  ratios  which  represent  the 
intervals  between  the  successive  notes. 

It  is  worthy  of  note  that  Pythagoras,  who  was  the  first  to  attempt 
the  numerical  evaluation  of  musical  intervals,  laid  down  a  scheme  of 
values  slightly  different  from  that  which  is  now  generally  adopted. 
According  to  him,  the  intervals  between  the  successive  notes  of  the 
major  scale  are  as  follows: — 

Do  Re  Mi  Fa  Sol  La  Si  Do 

i      t     IH     *      *      *     m 

This  scheme  agrees  exactly  with  the  common  system  as  regards  the 
values  of  the  fourth,  fifth,  and  octave,  and  makes  the  values  of  the 
major  third,  the  sixth,  and  the  seventh  each  greater  by  a  comma, 
while  the  small  interval  from  mi  to  fa,  or  from  si  to  do,  is  diminished 
by  a  comma.  In  the  ordinary  system,  the  prime  numbers  which 
enter  the  ratios  are  2,  3,  and  5 ;  in  the  Pythagorean  system  they  are 
only  2  and  3;  hence  the  interval  between  any  two  notes  of  the 
Pythagorean  scale  can  be  expressed  as  the  sum  or  difference  of  a 
certain  number  of  octaves  and  fifths.  In  tuning  a  violin  by  making 
the  intervals  between  the  strings  true  fifths,  the  Pythagorean  scheme 
is  virtually  employed. 


902 


NUMERICAL   EVALUATION   OF  SOUND. 


895.  Methods  of  Counting  Vibrations.  Siren. — The  instrument 
which  is  chiefly  employed  for  counting  the  number  of  vibrations 
corresponding  to  a  given  note,  is  called  the  siren,  and  was  devised 
by  Cagniard  de  Latour.  It  is  represented  in  Figs.  614,  615,  the 
former  being  a  front,  and  the  latter  a  back  view. 

There  is  a  small  wind-chest,  nearly  cylindrical,  having  its  top 
pierced  with  fifteen  holes,  disposed  at  equal  distances  round  the 
circumference  of  a  circle.  Just  over  this,  and  nearly  touching  it,  is 
a  movable  circular  plate,  pierced  with  the  same  number  of  holes 


Fig.  614. 


Siren. 


Fig.  615. 


similarly  arranged,  and  so  mounted  that  it  can  rotate  very  freely 
about  its  centre,  carrying  with  it  the  vertical  axis  to  which  it  is 
attached.  This  rotation  is  effected  by  the  action  of  the  wind,  which 
enters  the  wind-chest  from  below,  and  escapes  through  the  holes. 
The  form  of  the  holes  is  shown  by  the  section  in  Fig.  615.  They  do 
not  pass  perpendicularly  through  the  plates,  but  slope  contrary  ways, 
so  that  the  air  when  forced  through  the  holes  in  the  lower  plate 
impinges  upon  one  side  of  the  holes  in  the  upper  plate,  and  thus 
blows  it  round  in  a  definite  direction.  The  instrument  is  driven  by 
means  of  the  bellows  shown  in  Fig.  625  (§  910).  As  the  rotation  of 
one  plate  upon  the  other  causes  the  holes  to  be  alternately  opened 
and  closed,  the  wind  escapes  in  successive  puffs,  whose  frequency 


SIREN.  903 

depends  upon  the  rate  of  rotation.  Hence  a  note  is  emitted  which 
rises  in  pitch  as  the  rotation  becomes  more  rapid. 

The  siren  will  sound  under  water,  if  water  is  forced  through  it 
instead  of  air;  and  it  was  from  this  circumstance  that  it  derived  its 
name. 

In  each  revolution,  the  fifteen  holes  in  the  upper  plate  come 
opposite  to  those  in  the  lower  plate  15  times,  and  allow  the  com- 
pressed air  in  the  wind-chest  to  escape;  while  in  the  intervening 
positions  its  escape  is  almost  entirely  prevented.  Each  revolution 
thus  gives  rise  to  15  vibrations;  and  in  order  to  know  the  number 
of  vibrations  corresponding  to  the  note  emitted,  it  is  only  necessary 
to  have  a  means  of  counting  the  revolutions. 

This  is  furnished  by  a  counter,  which  is  represented  in  Fig.  615. 
The  revolving  axis  carries  an  endless  screw,  driving  a  wheel  of  100 
teeth,  whose  axis  carries  a  hand  traversing  a  dial  marked  with  100 
divisions.  Each  revolution  of  the  perforated  plate  causes  this  hand 
to  advance  one  division.  A  second  toothed-wheel  is  driven  inter- 
mittently by  the  first,  advancing  suddenly  one  tooth  whenever  the 
hand  belonging  to  the  first  wheel  passes  the  zero  of  its  scale.  This 
second  wheel  also  carries  a  hand  traversing  a  second  dial;  and  at 
each  of  the  sudden  movements  just  described  this  hand  advances  one 
division.  Each  division  accordingly  indicates  100  revolutions  of  the 
perforated  plate,  or  1500  vibrations.  By  pushing  in  one  of  the  two 
buttons  which  are  shown,  one  on  each  side  of  the  box  containing 
the  toothed- wheels,  we  can  instantaneously  connect  or  disconnect  the 
endless  screw  and  the  first  toothed- wheel. 

In  order  to  determine  the  number  of  vibrations  corresponding  to 
any  given  sound  which  we  have  the  power  of  maintaining  steadily, 
we  fix  the  siren  on  the  bellows,  the  screw  and  wheel  being  dis- 
connected, and  drive  the  siren  until  the  note  which  it  emits  is  judged 
to  be  in  unison  with  the  given  note.  We  then,  either  by  regulating 
the  pressure  of  the  wind,  or  by  employing  the  finger  to  press  with 
more  or  less  friction  against  the  revolving  axis,  contrive  to  keep  the 
note  of  the  siren  constant  for  a  measured  interval  of  time,  which  we 
observe  by  a  watch.  At  the  commencement  of  the  interval  we  sud- 
denly connect  the  screw  and  toothed-wheel,  and  at  its  termination 
we  suddenly  disconnect  them,  having  taken  care  to  keep  the  siren 
in  unison  with  the  given  sound  during  the  interval.  As  the  hands 
do  not  advance  on  the  dials  when  the  screw  is  out  of  connection  with 
the  wheels,  the  readings  before  and  after  the  measured  interval  of 


904  NUMERICAL   EVALUATION   OF  SOUND. 

time  can  be  taken  at  leisure.  Each  reading  consists  of  four  figures, 
indicating  the  number  of  revolutions  from  the  zero  position,  units 
and  tens  being  read  off  on  the  first  dial,  and  hundreds  and  thousands 
on  the  second.  The  difference  of  the  two  readings  is  the  number  of 
revolutions  made  in  the  measured  interval,  and  when  multiplied  by 
15  gives  the  number  of  vibrations  in  the  interval,  whence  the 
number  of  vibrations  per  second  is  computed  by  division. 

896.  Graphic  Method.— In  the  hands  of  a  skilful  operator,  with  a 
good  musical  ear,  the  siren  is  capable  of  yielding  very  accurate  deter- 
minations, especially  if,  by  adding  or  subtracting  the  number  of  beats, 


Fig.  616.— Vibroscope. 

correction  be  made  for  any  slight  difference  of  pitch  between  the 
siren  and  the  note  under  investigation. 

The  vibrations  of  a  tuning-fork  can  be  counted,  without  the  aid 
of  the  siren,  by  a  graphical  method,  which  does  not  call  for  any  exer- 
cise of  musical  judgment,  but  simply  involves  the  performance  of  a 
mechanical  operation. 

The  tuning-fork  is  fixed  in  a  horizontal  position,  as  shown  in  Fig. 
616,  and  has  a  light  style,  which  may  be  of  brass  wire,  quill,  or 
bristle,  attached  to  one  of  its  prongs,  by  wax  or  otherwise.  To 
receive  the  trace,  a  piece  of  smoked  paper  is  gummed  round  a 
cylinder,  which  can  be  turned  by  a  handle,  a  screw  cut  on  the  axis 


VIBROSCOPE   AND   PHONAUTOGRAPH.  905 

causing  it  at  the  same  time  to  travel  endwise.  The  cylinder  is 
placed  so  that  the  style  barely  touches  the  blackened  surface.  The 
fork  is  then  made  to  vibrate  by  bowing  it,  and  the  cylinder  is 
turned.  The  result  is  a  wavy  line  traced  on  the  blackened  surface, 
and  the  number  of  wave-forms  (each  including  a  pair  of  bends  in 
opposite  directions)  is  the  number  of  vibrations.  If  the  experiment 
lasts  for  a  measured  interval  of  time,  we  have  only  to  count  these 
wave-forms,  and  divide  by  the  number  of  seconds,  in  order  to  obtain 
the  number  of  vibrations  per  second  for  the  note  of  the  tuning-fork. 
By  plunging  the  paper  in  ether,  the  trace  will  be  fixed,  so  that  the 
paper  may  be  laid  aside,  and  the  vibrations  counted  at  leisure.  The 
apparatus  is  called  the  vibroscope,  and  was  invented  by  Duhamel. 

M.  Le'on  Scott  has  invented  an  instrument  called  the  phonauto- 
graph,  which  is  adapted  to  the  graphical  representation  of  sounds  in 


Fig.  617.— Traces  by  Phonaatograph. 

general.  The  style,  which  is  very  light,  is  attached  to  a  membrane 
stretched  across  the  smaller  end  of  what  may  be  called  a  large  ear- 
trumpet.  The  membrane  is  agitated  by  the  aerial  waves  proceeding 
from  any  source  of  sound,  and  the  style  leaves  a  record  of  these  agita- 
tions on  a  blackened  cylinder,  as  in  Duhamel's  apparatus.  Fig.  617 
represents  the  traces  thus  obtained  from  the  sound  of  a  tuning-fork 
in  three  different  modes  of  vibration. 

897.  Tonometer. — When  we  have  determined  the  frequency  of 
vibration  for  a  particular  tuning-fork,  that  of  another  fork,  nearly 
in  unison  with  it,  can  be  deduced  by  making  the  two  forks  vibrate 
simultaneously,  and  counting  the  beats  which  they  produce. 

Scheibler's  tonometer,  which  is  constructed  by  Koenig  of  Paris, 
consists  of  a  set  of  65  tuning-forks,  such  that  any  two  consecutive 
forks  make  4  beats  per  second,  and  consequently  differ  in  pitch  by 


906  NUMERICAL   EVALUATION   OF  SOUND. 

4  vibrations  per  second.  The  lowest  of  the  series  makes  256  vibra- 
tions, and  the  highest  512,  thus  completing  an  octave.  Any  note 
within  this  range  can  have  its  vibration-frequency  at  once  deter- 
mined, with  great  accuracy,  by  making  it  sound  simultaneously  with 
the  fork  next  above  or  below  it,  and  counting  beats. 

With  the  aid  of  this  instrument,  a  piano  can  be  tuned  with  cer- 
tainty to  any  desired  system  of  temperament,  by  first  tuning  the 
notes  which  come  within  the  compass  of  the  tonometer,  and  then 
proceeding  by  octaves. 

In  the  ordinary  methods  of  tuning  pianos  and  organs,  tempera- 
ment is  to  a  great  extent  a  matter  of  chance;  and  a  tuner  cannot 
attain  the  same  temperament  in  two  successive  attempts. 

898.  Pitch  modified  by  Relative  Motion.— We  have  stated  in  §  889 
that,  in  ordinary  circumstances,  the  frequency  of  vibration  in  the 
source  of  sound,  is  the  same  as  in  the  ear  of  the  listener,  and  in  the 
intervening  medium.  This  identity,  however,  does  not  hold  if  the 
source  of  sound  and  the  ear  of  the  listener  are  approaching  or  reced- 
ing from  each  other.  Approach  of  either  to  the  other  produces  in- 
creased frequency  of  the  pulses  on  the  ear,  and  consequent  elevation 
of  pitch  in  the  sound  as  heard;  while  recession  has  an  opposite  effect. 
Let  n  be  the  number  of  vibrations  performed  in  a  second  by  the 
source  of  the  sound,  v  the  velocity  of  sound  in  the  medium,  and  a 
the  relative  velocity  of  approach.  Then  the  number  of  waves  .vhich 
reach  the  ear  of  the  listener  in  a  second,  will  be  n  plus  the  number 
of  waves  which  cover  a  length  a,  that  is  (since  n  waves  cover  a 
length  v),  will  be  n+^n  orv-^n. 

The  following  investigation  is  more  rigorous.  Let  the  source 
make  n  vibrations  per  second.  Let  the  observer  move  towards  the 
source  with  velocity  a.  Let  the  source  move  away  from  the  observer 
with  velocity  a'.  Let  the  medium  move  from  the  observer  towards 
the  source  with  velocity  m,  and  let  the  velocity  of  sound  in  the 
medium  be  v. 

Then  the  velocity  of  the  observer  relative  to  the  medium  is  a  -  m 
towards  the  source,  and  the  velocity  of  the  source  relative  to  the 
medium  is  a'-m  away  from  the  observer.  The  velocity  of  the 
sound  relative  to  the  source  will  be  different  in  different  directions, 
its  greatest  amount  being  w+a'-m  towards  the  observer,  and  its 
least  being  -y-a'-fm  away  from  the  observer.  The  length  of  a 
wave  will  vary  with  direction,  being  \  of  the  velocity  of  the  sound 


PITCH   CHANGED   BY  APPROACH   OR  RECESS.  907 

relative  to  the  source.  The  length  of  those  waves  which  meet  the 
observer  will  be  —  — ,  and  the  velocity  of  these  waves  relative  to 
the  observer  will  be  v+a  ~m;  hence  the  number  of  waves  that  meet 
him  in  a  second  will  be  ^r™7i. 

Careful  observation  of  the  sound  of  a  railway  whistle,  as  an  express 
train  dashes  past  a  station,  has  confirmed  the  fact  that  the  sound  as 
heard  by  a  person  standing  at  the  station  is  higher  while  the  train 
is  approaching  than  when  it  is  receding.  A  speed  of  about  40  miles 
an  hour  will  sharpen  the  note  by  a  semitone  in  approaching,  and 
flatten  it  by  the  same  amount  in  receding,  the  natural  pitch  being 
heard  at  the  instant  of  passing.1 

1  The  best  observations  of  this  kind  were  those  of  Buys  Ballot,  in  which  trumpeters, 
with  their  instruments  previously  tuned  to  unison,  were  stationed,  one  on  the  locomotive, 
and  others  at  three  stations  beside  the  line  of  railway.  Each  trumpeter  was  accompanied 
by  musicians,  charged  with  the  duty  of  estimating  the  difference  of  pitch  between  the  note 
of  his  trumpet  and  those  of  the  others,  as  heard  before  and  after  passing. 


CHAPTER    LXIV. 


MODES   OF  VIBRATION. 


899.  Longitudinal  and  Transverse  Vibrations  of  Solids. — Sonorous 
vibrations  are  manifestations  of  elasticity.     When  the  particles  of  a 
solid  body  are  displaced  from  their  natural  positions  relative  to  one 
another  by  the  application  of  external  force,  they  tend  to  return,  in 
virtue  of  the  elasticity  of  the  body.     When  the  external  force  is 
removed,  they  spring  back  to  their  natural  position,  pass  it  in  virtue 
of  the  velocity  acquired  in  the  return,  and  execute  isochronous  vibra- 
tions about  it  until  they  gradually  come  to  rest.     The  isochronism 
of  the  vibrations  is  proved  by  the  constancy  of  pitch  of  the  sound 
emitted;  and  from  the  isochronism  we  can  infer,  by  the  aid  of  mathe- 
matical reasoning,  that  the  restoring  force  increases  directly  as  the 
displacement  of  the  parts  of  the  body  from  their  natural  relative 
position  (§  111). 

The  same  body  is,  in  general,  susceptible  of  many  different  modes 
of  vibration,  which  may  be  excited  by  applying  forces  to  it  in  dif- 
ferent ways.  The  most  important  of  these  are  comprehended  under 
the  two  heads  of  longitudinal  and  transverse  vibrations. 

In  the  former  the  particles  of  the  body  move  to  and  fro  in  the 
direction  along  which  the  pulses  travel,  which  is  always  regarded  as 
the  longitudinal  direction,  and  the  deformations  produced  consist  in 
alternate  compressions  and  extensions.  In  the  latter  the  particles 
move  to  and  fro  in  directions  transverse  to  that  in  which  the  pulses 
travel,  and  the  deformation  consists  in  bending.  To  produce  longi- 
tudinal vibrations,  we  must  apply  force  in  the  longitudinal  direction. 
To  produce  transverse  vibration,  we  must  apply  force  transversely. 

900.  Transverse  Vibrations  of  Strings. — To  the  transverse  vibra- 
tions of  strings,  instrumental  music  is  indebted  for  some  of  its  most 


VIBRATIONS   OF   STRINGS.  909 

precious  resources.  In  the  violin,  violoncello,  &c.,  the  strings  are  set 
in  vibration  by  drawing  a  bow  across  them.  The  part  of  the  bow 
which  acts  on  the  strings  consists  of  hairs  tightly  stretched  and 
rubbed  with  rosin.  The  bow  adheres  to  the  string,  and  draws  it 
aside  till  the  reaction  becomes  too  great  for  the  adhesion  to  overcome. 
As  the  bow  continues  to  be  drawn  on,  slipping  takes  place,  and  the 
mere  fact  of  slipping  diminishes  the  adhesion.  The  string  accordingly 
springs  back  suddenly  through  a  finite  distance.  It  is  then  again 
caught  by  the  bow,  and  the  same  action  is  repeated.  In  the 
harp  and  guitar,  the  strings  are  plucked  with  the  finger,  and  then  left 
to  vibrate  freely.  In  the  piano  the  wires  are  struck  with  little  ham- 
mers faced  with  leather.  The  pitch  of  the  sound  emitted  in  these 
various  cases  depends  only  on  the  string  itself,  and  is  the  same 
whichever  mode  of  excitation  be  employed. 

901.  Laws  of  the  Transverse  Vibrations  of  Strings.  —  It  can  be 
shown  by  an  investigation  closely  analogous  to  that  which  gives  the 
velocity  of  sound  in  air,  that  the  velocity  with  which  transverse 
vibrations  travel  along  a  perfectly  flexible  string  is  given  by  the 
formula 


•Vi- 


0) 


t  denoting  the  tension  of  the  string,  and  m  the  mass  of  unit  length 
of  it.  If  m  be  expressed  in  grammes  per  centimetre  of  length,  t 
should  be  expressed  in  dynes  (§  87),  and  the  value  obtained  for  v 
will  be  in  centimetres  per  second.  The  sudden  disturbance  of  any 
point  in  the  string,  causes  two  pulses  to  start  from  this  point,  and 
run  along  the  string  in  opposite  directions.  Each  of  these,  on 
arriving  at  the  end  of  the  free  portion  of -the  string,  is  reflected 
from  the  solid  support  to  which  the  string  is  attached,  and  at  the 
same  time  undergoes  reversal  as  to  side.  It  runs  back,  thus  reversed, 
to  the  other  end  of  the  free  portion,  and  there  again  undergoes 
reflection  and  reversal.  When  it  next  arrives  at  the  origin  of  the 
disturbance  it  has  travelled  over  just  twice  the  length  of  the  string; 
and  as  this  is  true  of  both  the  pulses,  they  must  both  arrive  at  this 
point  together.  At  the  instant  of  their  meeting,  things  are  in  the 
same  condition  as  when  the  pulses  were  originated,  and  the  move- 
ments just  described  will  again  take  place.  The  period  of  a  complete 
vibration  of  the  string  is  therefore  the  time  required  for  a  pulse  to 
travel  over  twice  its  length;  that  is, 


910  MODES  OF  VIBRATION. 

n=7=2'\/7; 

orn=¥l\/b  (2) 

I  denoting  the  length  of  the  string  between  its  points  of  attachment, 
and  n  the  number  of  vibrations  per  second. 
This  formula  involves  the  following  laws: — 

1.  When  the  length  of  the  vibrating  portion  of  the  string  is  altered, 
without  change  of  tension,  the  frequency  of  vibration  varies  inversely 
as  the  length. 

2.  If   the  tension  be  altered,  without  change  of   length  in  the 
vibrating  portion,  the  frequency  of  vibration  varies  as  the  square 
root  of  the  tension. 

3.  Strings  of  the  same  length,  stretched  with  the  same  forces,  have 
frequencies  of  vibration  which  are  inversely  as  the  square  roots  of 
their  masses  (or  weights). 

4.  Strings  of  the  same  length  and  density,  but  of  different  thick- 
nesses, will  vibrate  in  the  same  time,  if  they  are  stretched  with 
forces  proportional  to  their  sectional  areas. 

All  these  laws  are  illustrated  (qualitatively,  if  not  quantitatively) 
by  the  strings  of  a  violin. 

The  first  is  illustrated  by  the  fingering,  the  pitch  being  raised  as 
the  portion  of  string  between  the  finger  and  the  bridge  is  shortened. 

The  second  is  illustrated  by  the  mode  of  tuning,  which  consists  in 
tightening  the  string  if  its  pitch  is  to  be  raised,  or  slackening  the 
string  if  it  is  to  be  lowered. 

The  third  law  is  illustrated  by  the  construction  of  the  bass  string, 
which  is  wrapped  round  with  metal  wire,  for  the  purpose  of  adding  to 
its  mass,  and  thus  attaining  slow  vibration  without  undue  slackness. 
The  tension  of  this  string  is  in  fact  greater  than  that  of  the  string 
next  it,  though  the  latter  vibrates  more  rapidly  in  the  ratio  of  3  to  2. 

The  fourth  law  is  indirectly  illustrated  by  the  sizes  of  the  first 
three  strings.  The  treble  string  is  the  smallest,  and  is  nevertheless 
stretched  with  much  greater  force  than  any  of  the  others.  The  third 
string  is  the  thickest,  and  is  stretched  with  less  force  than  any  of 
the  others.  The  increased  thickness  is  necessary  in  order  to  give 
sufficient  power  in  spite  of  the  slackness  of  the  string. 

902.  Experimental  Illustration:  Sonometer. — For  the  quantitative 
illustration  of  these  laws,  the  instrument  called  the  sonometer,  re- 
presented in  Fig.  618,  is  commonly  employed.  It  consists  essen- 


VIBRATIONS   OF   STRINGS.  911 

tially  of  a  string  or  wire  stretched  over  a  sounding-box  by  means  of 
a  weight.  One  end  of  the  string  is  secured  to  a  fixed  point  at  one 
end  of  the  sounding-box.  The  other  end  passes  over  a  pulley,  and 
carries  weights  which  can  be  altered  at  pleasure.  Near  the  two 
ends  of  the  box  are  two  fixed  bridges,  over  which  the  cord  passes. 
There  is  also  a  movable  bridge,  which  can  be  employed  for  altering 
the  length  of  the  vibrating  portion. 

To  verify  the  law  of  lengths,  the  whole  length  between  the  fixed 
bridges  is  made  to  vibrate,  either  by  plucking  or  bowing;  the  mov- 


Fig.  618.— Sonometer. 

able  bridge  is  then  introduced  exactly  in  the  middle,  and  one  of  the 
halves  is  made  to  vibrate;  the  note  thus  obtained  will  be  found  to 
be  the  upper  octave  of  the  first.  The  frequency  of  vibration  is  there- 
fore doubled.  By  making  two-thirds  of  the  whole  length  vibrate, 
a  note  will  be  obtained  which  will  be  recognized  as  the  fifth  of  the 
fundamental  note,  its  vibration-frequency  being  therefore  greater  in 
the  ratio  f .  To  obtain  the  notes  of  the  gamut,  we  commence  with 
the  string  as  a  whole,  and  then  employ  portions  of  its  length  repre- 
sented by  the  fractions  f ,  %,  f ,  f ,  •§-,  -j^,  f . 

To  verify  the  law  independently  of  all  knowledge  of  musical  inter- 
vals, a  light  style  may  be  attached  to  the  cord,  and  caused  to  trace 
its  vibrations  on  the  vibroscope.  This  mode  of  proof  is  also  more 
geneval,  inasmuch  as  it  can  be  applied  to  ratios  which  do  not  corre- 
spond to  any  recognized  musical  interval. 

To  verify  the  law  of  tensions,  we  must  change  the  weight.  It 
will  be  found  that,  to  produce  a  rise  of  an  octave  in  pitch,  the  weight 
must  be  increased  fourfold. 

To  verify  the  third  and  fourth  laws,  two  strings  must  be  employed, 
their  masses  having  first  been  determined  by  weighing  them. 


912  MODES   OF  VIBRATION. 

If  the  strings  are  thick,  and  especially  if  they  are  thick  steel  wires, 
their  flexural  rigidity  has  a  sensible  effect  in  making  the  vibrations 
quicker  than  they  would  be  if  the  tension  acted  alone. 

903.  Harmonics. — Any  person  of  ordinary  musical  ear  may  easily, 
by  a  little  exercise  of  attention,  detect  in  any  note  of  a  piano  the 
presence  of  its  upper  octave,  and  of  another  note  a  fifth  higher  than 
this;  these  being  the  notes  which  correspond  to  frequencies  of  vibra- 
tion double  and  triple  that  of  the  fundamental  note.  A  highly 
trained  ear  can  detect  the  presence  of  other  notes,  corresponding  to 
still  higher  multiples  of  the  fundamental  frequency  of  vibration. 
Such  notes  are  called  harmonics. 

When  the  vibration-frequency  of  one  note  is  an  exact  'multiple 
of  that  of  another  note,  the  former  note  is  called  a  harmonic  of  the 
latter.  The  notes  of  all  stringed  instruments  contain  numerous 
harmonics  blended  with  the  fundamental  tones.  Bells  and  vibrating 
plates  have  higher  tones  mingled  with  the  fundamental  tone;  but 
these  higher  tones  are  not  harmonics  in  the  sense  in  which  we  use 
the  word. 

A  violin  string  sometimes  fails  to  yield  its  fundamental  note,  and 
gives  the  octave  or  some  other  harmonic  instead.  This  result  can  be 
brought  about  at  pleasure,  by  lightly  touching  the  string  at  a  pro- 
perly-selected point  in  its  length,  while  the  bow  is  applied  in  the 
usual  way.  If  touched  at  the  middle  point  of  its  length,  it  gives 
the  octave.  If  touched  at  one-third  of  its  length  from  either  end,  it 

gives  the  fifth  above  the  octave.     The  law  is,  that  if  touched  at  ^ 

of  its  length1  from  either  end,  it  yields  the  harmonic  whose  vibration- 
frequency  is  n  times  that  of  the  fundamental  tone.  The  string  in 
these  cases  divides  itself  into  a  number  of  equal  vibrating-segments, 
as  shown  in  Fig.  619. 

The  division  into  segments  is  often  distinctly  visible  when  the 
string  of  a  sonometer  is  strongly  bowed,  and  its  existence  can  be 
verified,  when  less  evident,  by  putting  paper  riders  on  different  parts 
of  the  string.  These  (as  shown  in  the  figure)  will  be  thrown  off  by 
the  vibrations  of  the  string,  unless  they  are  placed  accurately  at  the 
nodal  points,  in  which  case  they  will  retain  their  seats.  If  two 
strings  tuned  to  unison  are  stretched  on  the  same  sonometer,  the 
vibration  of  the  one  induces  similar  vibrations  in  the  other;  and  the 
experiment  of  the  riders  may  be  varied,  in  a  very  instructive  way, 

1  Or  at  ^  cf  its  length,  if  m  be  prime  to  n. 


SYMPATHETIC   VIBRATION   AND   RESONANCE.  913 

by  bowing  one  string  and  placing  the  riders  on  the  other.  This  is 
an  instance  of  a  general  principle  of  great  importance — that  a  vi- 
brating body  communicates  its  vibrations  to  other  bodies  which  are 
capable  of  vibrating  in  unison  with  it.  The  propagation  of  a  sound 
may  indeed  be  regarded  as  one  grand  vibration  in  unison;  but, 
besides  the  general  waves  of  propagation,  there  are  waves  of  re- 


Fig,  619. — Production  of  a  Harmonic. 

inforcement,  due  to  the  synchronous  vibrations  of  limited  portions 
of  the  transmitting  medium.  This  is  the  principle  of  resonance. 

904.  Resonance. — By  applying  to  a  pendulum  originally  at  rest 
a  series  of  very  feeble  impulses,  at  intervals  precisely  equal  to  its 
natural  time  of  vibration,  we  shall  cause  it  to  swing  through  an  arc 
of  considerable  magnitude. 

The  same  principle  applies  to  a  body  capable  of  executing  vibra- 
tions under  the  influence  of  its  own  elasticity.  A  series  of  impulses 
keeping  time  with  its  own  natural  period  may  set  it  in  powerful 
vibration,  though  any  one  of  them  singly  would  have  no  appreciable 
effect. 

Some  bodies,  such  as  strings  and  confined  portions  of  air,  have 
definite  periods  in  which  they  can  vibrate  freely  when  once  started; 
68 


914  MODES   OF  VIBRATION. 

and  when  a  note  corresponding  to  one  of  these  periods  is  sounded  in 
their  neighbourhood,  they  readily  take  it  up  and  emit  a  note  of  the 
same  pitch  themselves. 

Other  bodies,  especially  thin  pieces  of  dry  straight-grained  deal, 
such  as  are  employed  for  the  faces  of  violins  and  the  sounding- 
boards  of  pianos,  are  capable  of  vibrating,  more  or  less  freely,  in  any 
period  lying  between  certain  wide  limits.  They  are  accordingly  set 
in  vibration  by  all  the  notes  of  their  respective  instruments;  and  by 
the  large  surface  with  which  they  act  upon  the  air,  they  contribute 
in  a  very  high  degree  to  increase  the  sonorous  effect.  All  stringed 
instruments  are  constructed  on  this  principle;  and  their  quality 
mainly  depends  on  the  greater  or  less  readiness  with  which  they 
respond  to  the  vibrations  of  the  strings. 

All  such  methods  of  reinforcing  a  sound  must  be  included  under 
resonance;  but  the  word  is  often  more  particularly  applied  to  the 
reinforcement  produced  by  masses  of  air. 

905.  Longitudinal  Vibrations  of  Strings. — Strings  or  wires  may 
also  be  made  to  vibrate  longitudinally,  by  rubbing  them,  in  the 
direction  of  their  length,  with  a  bow  or  a  piece  of  chamois  leather 
covered  with  rosin.  The  sounds  thus  obtained  are  of  much  higher 
pitch  than  those  produced  by  transverse  vibration. 

In  the  case  of  the  fundamental  note,  each  of  the  two  halves  A  C, 
C  B  (Fig.  620),  is  alternately  extended  and  compressed,  one  being 


Fig.  620.— Longitudinal  Vibration.    First  Tone. 

extended  while  the  other  is  compressed.  At  the  middle  point  C 
there  is  no  extension  or  compression,  but  there  is  greater  amplitude 
of  movement  than  at  any  other  point.  The  amplitudes  diminish  in 
passing  from  C  towards  either  end,  and  vanish  at  the  ends,  which 
are  therefore  nodes.  The  extensions  and  compressions,  on  the  other 
hand,  increase  as  we  travel  from  the  middle  towards  either  end,  and 
obtain  their  greatest  values  at  the  ends. 

But  the  string  may  also  divide  itself  into  any  number  of  separately- 
vibrating  segments,  just  as  in  the  case  of  transverse  vibrations. 
Fig.  621  represents  the  motions  which  occur  when  there  are  three 
such  segments,  separated  by  two  nodes  D,  E.  The  upper  portion  of 
the  figure  is  true  for  one-half  of  the  period  of  vibration,  and  the  lower 
portion  for  the  remaining  half. 


CHLADNI'S   FIGURES.  915 

The  frequency  of  vibration,  for  longitudinal  as  well  as  for  trans- 
verse vibrations,  varies  inversely  as  the  length  of  the  vibrating 
string,  or  segment  of  string.  We  shall  return  to  this  subject  in  §  916. 


Fig.  621.— Longitudinal  Vibration.    Third  Tone. 

908.  Stringed  Instruments. — Only  the  transversal  vibrations  of 
strings  are  employed  in  music.  In  the  violin  and  violoncello  there 
are  four  strings,  each  being  tuned  a  fifth  above  the  next  below  it; 
and  intermediate  notes  are  obtained  by  fingering,  the  portion  of 
string  between  the  finger  and  the  bridge  being  the  only  part  that  is 
free  to  vibrate.  The  bridge  and  sounding-post  serve  to  transmit  the 
vibrations  of  the  strings  to  the  body  of  the  instrument.  In  the  piano 
there  is  also  a  bridge,  which  is  attached  to  the  sounding-board,  and 
communicates  to  it  the  vibrations  of  the  wires. 

907.  Transversal  Vibrations  of  Rigid  Bodies:  Rods,  Plates,  Bells.— 
We  shall  not  enter  into  detail  respecting  the  laws  of  the  transverse 
vibrations  of  rigid  bodies.  The  relations  of  their  overtones  to  their 
fundamental  tones  are  usually  of  an  extremely  complex  character, 
and  this  fact  is  closely  connected  with  the  unmusical  or  only  semi- 
musical  character  of  the  sounds  emitted. 

When  one  face  of  the  body  is  horizontal,  the  division  into  separate 
vibrating  segments  can  be  rendered  visible  by  a  method  devised  by 
Chladni,  namely,  by  strewing  sand  on  this  face.  During  the  vibra- 
tion, the  sand,  as  it  is  tossed  about,  works  its  way  to  certain  definite 
lines,  where  it  comes  nearly  to  rest.  These  nodal  lines  must  be 
regarded  as  the  intersections  of  internal  nodal  surfaces  with  the 
surface  on  which  the  sand  is  strewed,  each  nodal  surface  being  the 
boundary  between  parts  of  the  body  which  have  opposite  motions. 

The  figures  composed  by  these  nodal  lines  are  often  very  beautiful, 
and  quite  startling  in  the  suddenness  of  their  production.  Chladni 
and  Savart  published  the  forms  of  a  great  number.  A  complete 
theoretical  explanation  of  them  would  probably  transcend  the  powers 
of  the  greatest  mathematicians. 

Bells  and  bell-glasses  vibrate  in  segments,  which  are  never  less 
than  four  in  number,  and  are  separated  by  nodal  lines  meeting  in  the 
middle  of  the  crown.  They  are  well  shown  by  putting  water  in  a 


916  MODES   OF   VIBRATION. 

bell-glass,  and  bowing  its  edge.  The  surface  of  the  water  will  im- 
mediately be  covered  with  ripples,  one  set  of  ripples  proceeding  from 
each  of  the  vibrating  segments.  The  division  into  any  possible 
number  of  segments  may  be  effected  by  pressing  the  glass  with  the 
fingers  in  the  places  where  a  pair  of  consecutive  nodes  ought  to  be 
formed,  while  the  bow  is  applied  to  the  middle  of  one  of  the  seg- 
ments. The  greater  the  number  of  segments  the  higher  will  be  the 
note  emitted. 

908.  Tuning-fork. — Steel  rods,  on  account  of  their  comparative 
freedom  from  change,  are  well  suited  for  standards  of  pitch.     The 
tuning-fork,  which  is  especially  used  for  this  purpose,  consists  essen- 
tially of  a  steel  rod  bent  double,  and  attached  to  a  handle  of  the  same 
material  at  its  centre.     Besides  the  fundamental  tone,  it  is  capable 
of  yielding  two  or  three  overtones,  which  are  very  much  higher  in 
pitch;  but  these  are  never  used  for  musical  purposes.     If  the  fork 
is  held  by  the  handle  while  vibrating,  its  motion  continues  for  a 
long  time,  but  the  sound  emitted  is  too  faint  to  be  heard  except 

by  holding  the  ear  near  it.  When  the 
handle  is  pressed  against  a  table,  the 
latter  acts  as  a  sounding-board,  and 
communicates  the  vibrations  to  the 
air,  but  it  also  causes  the  fork  to 
come  much  more  speedily  to  rest.  For 
the  purposes  of  the  lecture-room  the 
fork  is  often  mounted  on  a  sounding- 
box  (Fig.  622),  which  should  be  sepa- 

622. -Fork  on  Sounding-box.        rated  from  the  table  by  two  pieces  of 
india-rubber  tubing.      The   box  can 

then  vibrate  freely  in  unison  with  the  fork,  and  the  sound  is  both 
loud  and  lasting.  The  vibrations  are  usually  excited  either  by  bow- 
ing the  fork  or  by  drawing  a  piece  of  wood  between  its  prongs. 

The  pitch  of  a  tuning-fork  varies  slightly  with  temperature,  be- 
coming lower  as  the  temperature  rises.  This  effect  is  due  in  some 
trifling  degree  to  expansion,  but  much  more  to  the  diminution  of 
elastic  force. 

909.  Law  of  Linear  Dimensions. — The  following  law  is  of  very  wide 
application,  being  applicable  alike  to  solid,  liquid,  and  gaseous  bodies: 
—  When  two  bodies  differing  in  size,  but  in  other  respects  similar 
and  similarly  circumstanced,  vibrate  in  the  same  mode,  their  vibra- 
tion-periods are  directly  as  their  linear  dimensions.     Their  vibra- 


ORGAN   PIPES. 


917 


tion-frequencies  are  consequently  in  the  inverse  ratio  of  their  linear 
dimensions. 

In  applying  the  law  to  the  transverse  vibrations  of  strings,  it  is 
to  be  understood  that  the  stretching  force  per  unit  of  sectional  area 
is  constant.  In  this  case  the  velocity  of  a  pulse  (§  901)  is  constant, 
and  the  period  of  vibration,  being  the  time  required  for  a  pulse  to 
travel  over  twice  the  length  of  the  string,  is  therefore  directly  as  the 
length. 

910.  Organ-pipes. — In  organs,  and  wind-instruments  generally,  the 
sonorous  body  is  a  column  of  air  confined  in  a  tube.  To  set  this  air 


Fig.  623.-Block  Pipe. 


Fig.  624. -Flue  Pipe. 


in  vibration  some  kind  of  mouth-piece  must  be  employed.  That 
which  is  most  extensively  used  in  organs  is  called  the  flute  mouth- 
piece? and  is  represented,  in  conjunction  with  the  pipe  to  which  it  is 
attached,  in  Figs.  623,  G24.  It  closely  resembles  the  mouth-piece  of 


1  This  is  not  the  trade  name, 
mouth-piece. 


English  organ-builders  have  no  generic  name  for  this 


918 


MODES   OF  VIBRATION. 


an  ordinary  whistle.     The  air  from  the  bellows  arrives  through  the 
conical  tube  at  the  lower  end,  and,  escaping  through  a  narrow  slit, 

grazes  the  edge  of  a 
wedge  placed  opposite. 
A  rushing  noise  is  thus 
produced,  which  con- 
tains, among  its  consti- 
tuents, the  note  to  which 
the  column  of  air  in  the 
pipe  is  capable  of  re- 
sounding; and  as  soon  as 
this  resonance  occurs,  the 
pipe  speaks.  Fig.  623 
represents  a  wooden  and 
Fig.  624  a  metal  organ- 
pipe,  both  of  them  being 
furnished  with  flute 
mouth-pieces.  The  two 
arrows  in  the  sections 
are  intended  to  suggest 
the  two  courses  which 
the  wind  may  take  as 
it  issues  from  the  slit, 
one  of  which  it  actually 
selects  to  the  exclusion 
of  the  other. 

The  arrangements  for 
admitting  the  wind  to 
the  pipes  by  putting 
down  the  keys  are 
shown  in  Fig.  625.  The 
bellows  V  are  worked 
by  the  treadle  P.  The  force  of  the  blast  can  be  increased  by  weight- 
ing the  top  of  the  bellows,  or  by  pressing  on  the  rod  T.  The  air 
passes  up  from  the  bellows,  through  a  large  tube  shown  at  one  end, 
into  a  reservoir  C,  called  the  wind-chest.  In  the  top  of  the  wind- 
chest  there  are  numerous  openings  c,  d,  &c.,  in  which  the  tubes  are 
to  be  fixed.  The  sectional  drawing  in  the  upper  part  of  the  figure 
shows  the  internal  communications.  A  plate  K,  pressed  up  by  a 
spring  R,  cuts  off  the  tube  c  from  the  wind-chest,  until  the  pin  a 


il  Organ. 


ORGAN  PIPES. 


919 


is  depressed.  The  putting  down  of  this  pin  lowers  the  plate,  and 
admits  the  wind.  This  description  only  applies  to  the  experimental 
organs  which  are  constructed  for  lecture  illustration.  In  real  organs 
the  pressure  of  the  wind  in  the  bellows  is  constant;  and  as  this 
pressure  would  be  too  great  for  most  of  the  pipes,  the  several  aper- 
tures of  admission  are  partially  plugged,  to  diminish  the  force  of  the 
blast. 

911.  The  Air  is  the  Sonorous  Body.  —  It  is  easily  shown  that  the 
sound  emitted  by  an  organ-pipe  depends,  mainly  at  least,  on  the 
dimensions  of  the  inclosed  column  of  air.  and  not  on  the  thickness 
or  material  of  the  pipe  itself.     For  let  three  pipes,  one  of  wood,  one 
of  copper,  and  the  other  of  thick  card,  all  of   the  same  internal 
dimensions,  be  fixed  on  the  wind-chest.     On  making  them  speak,  it 
will  be  found  that  the  three  sounds  have  exactly  the  same  pitch,  and 
but  slight  difference  in  character.     If,  however,  the  sides  of  the  tube 
are  excessively  thin,  their  yielding  has  a  sensible  influence,  and  the 
pitch  of  the  sound  is  modified. 

912.  Law  of  Linear  Dimensions.  —  The  law  of  linear  dimensions, 
stated  in  §  909  as  applying  to  the  vibrations  of  similar  solid  bodies, 
applies  to  gases  also.    Let 

two     box  -shaped     pipes 

(Fig.    626)    of    precisely 

similar  form,  and  having 

their  linear  dimensions  in 

the  ratio  of  2  :  1,  be  fixed 

on  the  wind-chest;  it  will 

be  found,  on  making  them 

speak,  that  the  note  of  the 

small   one   is   an  octave 

higher  than  the  other;  — 

showingdouble  frequency  rig 

of  vibration. 

913.  Bernoulli's  Laws.  —  The  law  just  stated  applies  to  the  com- 
parison of  similar  tubes  of  any  shape  whatever.     When  the  length 
of  a  tube  is  a  large  multiple  of  its  diameter,  the  note  emitted  is 
nearly  independent  of  the  diameter,  and  depends  almost  entirely  on 
the  length.     The  relations  between  the  fundamental  note  of  such  a 
tube  and  its  overtones  were  discovered  by  Daniel  Bernoulli,  and  are 
as  follows:  — 

I.  Overtones  of  Open  Pipes—  Let  the  pipe  B  (Fig.  627),  which  is 


_^  of  Linear  Dimens 


920 


MODES   OF   VIBKATION. 


open  at  the  upper  end,  be  fixed  on  the  wind-chest;  let  the  correspond- 
ing key  be  put  down,  and  the  wind  gradually  turned  on,  by  means 
of  the  cock  below  the  mouth-piece.  The  first  note  heard 
will  be  feeble  and  deep;  it  is  the  fundamental  note  of  the 
pipe.  As  the  wind  is  gradually  turned  full  on,  and  in- 
creasing pressure  afterwards  applied  to  the  bellows,  a 
series  of  notes  will  be  heard,  each  higher  than  its  pre- 
decessor. These  are  the  overtones  of  the  pipe.  They  are 
the  harmonics  of  the  fundamental  note;  that  is  to  say,  if 
1  denote  the  frequency  of  vibration  for  the  fundamental 
tone,  the  frequencies  of  vibration  for  the  overtones  will  be 
approximately  2,  3,  4,  5  ...  respectively. 

II.  Overtones  of  Stopped  Pipes. — If  the  same  experi- 
ment be  tried  with  the  pipe  A,  which  is  closed  at  its 
upper  end;  the  overtones  will  form  the  series  of  odd 
harmonics  of  the  fundamental  note,  all  the  even  har- 
monics being  absent;  in  other  words,  the  frequencies  of 
vibration  of  the  fundamental  tone  and  overtones  will  be 
approximately  represented  by  the  series  of  odd  numbers 
1,  3,  5,  7  ... 

rig.  627.  It  will  also  be  found,  that  if  both  pipes  are  of  the  same 
for  overtones  length,  tne  fundamental  note  of  the  stopped  pipe  is  an 

octave  lower  than  that  of  the  open  pipe. 

914.  Mode  of  Production  of  Overtones. — In  the  production  of  the 
overtones,  the  column  of  air  in  a  pipe  divides  itself  into  vibrating 
segments,  separated  by  nodal  cross-sections.  At  equal  distances  on 
opposite  sides  of  a  node,  the  particles  of  air  have  always  equal  and 
opposite  velocities,  so  that  the  air  at  the  node  is  always  subjected  to 
equal  forces  in  opposite  directions,  and  thus  remains  unmoved  by 
their  action.  The  portion  of  air  constituting  a  vibrating  segment, 
sways  alternately  in  opposite  directions,  and  as  the  movements  in 
two  consecutive  segments  are  opposite,  two  consecutive  nodes  are 
always  in  opposite  conditions  as  regards  compression  and  extension. 
The  middle  of  a  vibrating  segment  is  the  place  where  the  ampli- 
tude of  vibration  is  greatest,  and  the  variation  of  density  least. 
It  may  be  called  an  antinode.  The  distance  from  one  node  to  the 
next  is  half  a  wave-length,  and  the  distance  from  a  node  to  an  anti- 
node  is  a  quarter  of  a  wave-length.  Both  ends  of  an  open  pipe,  and 
the  end  next  the  mouth-piece  of  a  stopped  pipe,  are  antinodes,  being 
preserved  from  changes  of  density  by  their  free  communication  with 


STATIONARY   UNDULATION.  921 

the  external  air.  At  the  closed  end  of  a  stopped  pipe  there  must 
always  be  a  node. 

The  swaying  to  and  fro  of  the  internodal  portions  of  air  between 
fixed  nodal  planes,  is  an  example  of  stationary  undulation;  and  the 
vibration  of  a  musical  string  is  another  example.  A  stationary 
undulation  may  always  be  analysed  into  two  component  undulations 
equal  and  similar  to  one  another,  and  travelling  in  opposite  direc- 
tions, their  common  wave-length  being  double  of  the  distance  from 
node  to  node  •(§  887).  These  undulations  are  constantly  undergoing 
reflection  from  the  ends  of  the  pipe  or  string,  and,  in  the  case  of  pipes, 
the  reflection  is  opposite  in  kind  according  as  it  takes  place  from  a 
closed  or  an  open  end.  In  the  former  case  a  condensation  propagated 
towards  the  end  is  reflected  as  a  condensation,  the  forward-moving 
particles  being  compelled  to  recoil  by  the  resistance  which  they  there 
encounter;  and  a  rarefaction  is,  in  like  manner,  reflected  as  a  rare- 
faction. On  the  other  hand,  when  a  condensation  arrives  at  an  open 
end,  the  sudden  opportunity  for  expansion  which  is  afforded  causes 
an  outward  movement  in  excess  of  that  which  would  suffice  for 
equilibrium  of  pressure,  and  a  rarefaction  is  thus  produced  which  is 
propagated  back  through  the  tube.  A  condensation  is  thus  reflected 
as  a  rarefaction;  and  a  rarefaction  is,  in  like  manner,  reflected  as  a 
condensation. 

The  period  of  vibration  of  the  fundamental  note  of  a  stopped  pipe 
is  the  time  required  for  propagating  a  pulse  through  four  times  the 
length  of  the  pipe.  For  let  a  condensation  be  suddenly  produced  at 
the  lower  end  by  the  action  of  the  vibrating  lip.  It  will  be  pro- 
pagated to  the  closed  end  and  reflected  back,  thus  travelling  over 
twice  the  length  of  the  pipe.  On  arriving  at  the  aperture  where  the 
lip  is  situated,  it  is  reflected  as  a  rarefaction.  This  rarefaction  travels 
to  the  closed  end  and  back,  as  the  condensation  did  before  it,  and  is 
then  reflected  from  the  aperture  as  a  condensation.  Things  are  now 
in  their  initial  condition,  and  one  complete  vibration  has  been  per- 
formed. The  period  of  the  movements  of  the  lip  is  determined  by 
the  arrival  of  these  alternate  condensations  and  rarefactions ;  and  the 
lip,  in  its  turn,  serves  to  divert  a  portion  of  the  energy  of  the  blast, 
and  employ  it  in  maintaining  the  energy  of  the  vibrating  column. 

The  wave-length  of  the  fundamental  note  of  a  stopped  pipe  is  thus 
four  times  the  length  of  the  pipe. 

In  an  open  pipe,  a  condensation,  starting  from  the  mouth-piece,  is 
reflected  from  the  other  end  as  a  rarefaction.  This  rarefaction,  on 


922  MODES   OF  VIBRATION. 

reaching  the  mouth-piece,  is  reflected  as  a  condensation;  and  things 
are  thus  in  their  initial  state  after  the  length  of  the  pipe  has  been 
traversed  twice.  The  period  of  vibration  of  the  fundamental  note  is 
accordingly  the  time  of  travelling  over  twice  the  length  of  the  pipe; 
and  its  wave-length  is  twice  the  length  of  the  pipe.  In  every  case 
of  longitudinal  vibration,  if  the  reflection  is  alike  at  both  ends,  the 
wave-length  of  the  fundamental  tone  is  twice  the  distance  between 
the  ends. 

915.  Explanation  of  Barnoulli's  Laws. —  In  investigating  the 
theoretical  relations  between  the  fundamental  tone  and  overtones  for 
a  pipe  of  either  kind,  it  is  convenient  to  bear  in  mind  that  the  dis- 
tance from  an  open  end  to  the  nearest  node  is  a  quarter  of  a  wave- 
length of  the  note  emitted. 

In  the  case  of  the  open  pipe  the  first  or  fundamental  tone  requires 
one  node,  which  is  at  the  middle  of  the  length.  The  second  tone 
requires  two  nodes,  with  half  a  wave-length  between  them,  while 
each  of  them  is  a  quarter  of  a  wave-length  from  the  nearest  end.  A 
quarter  wave-length  has  thus  only  half  the  length  which  it  had  for 
the  fundamental  tone,  and  the  frequency  of  vibration  is  therefore 
doubled. 

The  third  tone  requires  three  nodes,  and  the  distance  from  either  end 
to  the  nearest  node  is  %  of  the  length  of  the  pipe,  instead  of  £  the 
length  as  in  the  case  of  the  first  tone.  The  wave-length  is  thus 
divided  by  3,  and  the  frequency  of  vibration  is  increased  threefold. 
We  can  evidently  account  in  this  way  for  the  production  of  the 
complete  series  of  harmonics  of  the  fundamental  note. 

In  the  case  of  the  stopped  pipe,  the  mouth-piece  is  always  distant 
a  quarter  wave-length  from  the  nearest  node,  and  this  must  be  dis- 
tant an  even  number  of  quarter  wave-lengths  from  the  stopped  end, 
which  is  itself  a  node. 

For  the  fundamental  tone,  a  quarter  wave-length  is  the  whole 
length  of  the  pipe. 

For  the  second  tone,  there  is  one  node  besides  that  at  the  closed 
end,  and  its  distance  from  the  open  end  is  £  of  the  length  of  the  pipe. 

For  the  third  tone,  there  are  two  nodes  besides  that  at  the  closed 
end.  The  distance  from  the  open  end  to  the  nearest  node  is  there- 
fore £  of  the  length  of  the  pipe. 

The  wave-lengths  of  the  successive  tones,  beginning  with  the 
fundamental,  are  therefore  as  1,  £,£,•}...,  and  their  vibration- 
frequencies  are  as  1,  3,  5,  7  ... 


RODS  AND   STRINGS.  923 

Also,  since  the  wave-length  of  the  fundamental  tone  is  four  times 
the  length  of  the  pipe  if  stopped,  and  only  twice  its  length  if  open,  it 
is  obvious  that  the  wave-length  is  halved,  and  the  frequency 
of  vibration  doubled,  by  unstopping  the  pipe. 

No  change  of  pitch,  or  only  very  slight  change,  will  be 
produced  by  inserting  a  solid  partition  at  a  node,  or  by  put- 
ting an  antinode  in  free  communication  with  the  external  air. 
These  principles  can  be  illustrated  by  means  of  the  jointed 
pipe  represented  in  Fig.  628. 

916.  Application  to  Bods  and  Strings. — The  same  laws 
which  apply  to  open  organ-pipes,  also  apply  to  the  longi- 
tudinal vibrations  of  rods  free  at  both  ends,  and  to  both  the 
longitudinal  and  transverse  vibrations  of  strings.   In  all  these 
cases  the  overtones  form  the  complete  series  of  harmonics  of 
the  first  or  fundamental  tone,  and  the  period  of  vibration 
for  this  first  tone  is  the  time  occupied  by  a  pulse  in  travel- 
ling over  twice  the  length  of  the  given  rod  or  string.     In 
the  case  of  longitudinal  vibrations  the  velocity  of  a  pulse  is 

A/ j),  M  denoting  the  value  of  Young's  modulus  for  the 
rod  or  string,  and  D  its  density.  This  is  identical  with 
the  velocity  of  sound  through  the  rod  or  string,  and  is 

i       ,      c    -L  •  TO  c   L  i          Jointed 

independent  of  its  tension.     In  the  case  01  transverse  pulses    pjpe. 
in  a  string  (regarded  as  perfectly  flexible),  the  formula  for  the 

/-p 

velocity  of  transmission  (1)  §  901,  may  be  written  A/JJ.  F  denot- 
ing the  stretching  force  per  unit  of  sectional  area.  The  ratio  of  the 
latter  velocity  to  the  former  is  A/^,  which  is  always  a  small  frac- 
tion, since  ^  expresses  the  fraction  of  itself  by  which  the  string  is 

lengthened  by  the  force  F. 

If  a  rod,  free  at  both  ends,  is  made  to  vibrate  longitudinally,  its 
nodes  and  antinodes  will  be  distributed  exactly  in  the  same  way  as 
those  of  an  open  organ-pipe.  The  experiment  can  be  performed  by 
holding  the  rod  at  a  node,  and  rubbing  it  with  rosined  chamois 
leather. 

917.  Application  to  Measurement  of  Velocity  in  Gases. — Let  v  denote 
the  velocity  of  sound  in  a  particular  gas,  in  feet  per  second,  X  the 
wave-length  of  a  particular  note  in  this  gas  in  feet,  and  n  the  fre- 
quency of  vibration  for  this  note,  that  is  the  number  of  vibrations 


924 


MODES   OF   VIBRATION. 


per  second  which  produce  it.     Then  \  is  the  distance  travelled  in  — 
of  a  second,  and  the  distance  travelled  in  a  second  is 

v  —  n\. 

For  the  same  note,  n  is  constant  for  all  media  whatever,  and  v  varies 
directly  as  \.  The  velocities  of  sound  in  two  gases  may  thus  be 
compared  by  observing  the  lengths  of  vibrating  columns  of  the  two 
gases  which  give  the  same  note;  or  if  columns  of  equal  length  be 
employed,  the  velocities  will  be  directly  as  the  frequencies  of  vibra- 
tion, which  are  determined  by  observing  the  pitch  of  the  notes 
emitted. 

By  these  methods,  Dulong,  and  more  recently  Wer- 
theim,  have  determined  the  velocity  of  sound  in  several 
different  gases.  The  following  are  Wertheim's  results,  in 
metres  per  second,  the  gases  being  supposed  to  be  at  0°  C. 


Air, 

Oxygen,      .     .     . 
Hydrogen, 
Carbonic  oxide,  . 


331 

317 

1269 

337 


Carbonic  acid, 
Nitrous  oxide, 
Olefiant  gas, 


262 
262 
314 


The  same  principle  is 
applicable  to  liquids 
and  solids;  and  it 
was  by  means  of  the 
longitudinal  vibra- 
tions of  rods  that 
the  velocities  given 
in  §  880  were  ascer- 
tained. 

918.  Reed-pipes.— 
Instead  of  the  flute 
mouth -piece  above 
described,organ-pipes 
are  often  furnished 
with  what  is  called 
a  reed.  A  reed  con- 
tains an  elastic  plate 
I  (Figs.  629,  630)  call- 
ed the  tongue,  which, 
by  its  vibrations,  al- 
ternately opens  and  closes  or  nearly  closes  an  aperture  through  which 
the  wind  passes.  In  Fig.  629,  the  air  from  the  bellows  enters  first 


Fig.  629. -Heed  Pipe. 


Fig.  630. -Free  Reed. 


REED   PIPES.  925 

the  lower  part  t  of  the  pipe,  and  thence  (when  permitted  by  the 
tongue)  passes  through  the  channel l  r  into  the  upper  part  t'.  The 
stiff  wire  z,  movable  with  considerable  friction  through  the  hole  b, 
limits  the  vibrating  portion  of  the  tongue,  and  is  employed  for 
tuning.  Reed-pipes  are  often  terminated  above  by  a  trumpet-shaped 
expansion. 

A  striking  reed  (Fig.  629)  is  one  whose  tongue  closes  the  aperture 
by  covering  it.  The  tongue  should  be  so  shaped  as  not  to  strike 
along  its  whole  length  at  once,  but  to  roll  itself  down  over  the  aper- 
ture. In  the  free  reed  (Fig.  630)  the  tongue  can  pass  completely 
through. 

The  striking  reed  is  generally  preferred  in  organs,  its  peculiar 
character  rendering  it  very  effective  by  way  of  contrast.  It  is  always 
used  for  the  trumpet  stop.  Reed-pipes  can  be  very  strongly  blown 
without  breaking  into  overtones.  Their  pitch,  however,  if  they  are 
of  the  striking  kind,  is  not  independent  of  the  pressure  of  the  wind, 
but  gradually  rises  as  the  pressure  increases.  Free  reeds,  which  are 
used  for  harmoniums,  accordions,  and  concertinas,  do  not  change  in 
pitch  with  change  of  pressure. 

Elevation  of  temperature  sharpens  pipes  with  flute  mouth-pieces, 
and  flattens  reed-pipes.  The  sharpening  is  due  to  the  increased  velo- 
city of  sound  in  hot  air.  The  flattening  is  due  to  the  diminished 
elasticity  of  the  metal  tongue.  It  is  thus  proved  that  the  pitch  of  a 
reed-pipe  is  not  always  that  due  to  the  free  vibration  of  the  inclosed 
air,  but  may  be  modified  by  the  action  of  the  tongue. 

919.  Wind-instruments. — In  all  wind-instruments,  the  sound  is 
originated  by  one  of  the  two  methods  just  described.  With  the  flute' 
pipe  must  be  classed  the  flute,  the  flageolet,  and  the  Pandean-pipes. 
The  clarionet,  hautboy,  and  bassoon  have  reed  mouth-pieces,  the 
vibrating  tongue  being  a  piece  of  reed  or  cane.  In  the  bugle,  trum- 
pet, and  French-horn,  which  are  mere  tubes  without  keys,  the  lips 
of  the  performer  act  as  the  reed-tongue,  and  the  notes  produced  are 
approximately  the  natural  overtones.  These,  when  of  high  order,  are 
so  near  together,  that  a  gamut  can  be  formed  by  properly  selecting 
from  among  them. 

The  fingering  of  the  flute  and  clarionet,  has  the  effect  sometimes 
of  altering  the  effective  length  of  the  vibrating  column  of  air,  and 
sometimes  of  determining  the  production  of  overtones.  In  the 

1  The  piece  r,  which  is  approximately  a  half  cylinder,  is  called  the  reed  by  organ- 
builders. 


926 


MODES   OF  VIBRATION. 


trombone  and  cornet-a-piston,  the  length  of  the  vibrating  column 
of  air  is  altered.  The  harmonium,  accordion,  and  concertina  are 
reed  instruments,  the  reeds  employed  being  always  of  the  free 
kind. 

920.  Manometric  Flames. — Koenig,  of  Paris,  constructs  several 
forms  of  apparatus,  in  which  the  varia- 
tions of  pressure  produced  by  vibrations 
of  air  in  a  pipe  are  rendered  evident  to 
the  eye  by  their  effect  upon  flames. 
One  of  these  is  represented  in  Fig.  631. 
Three  small  gas-burners  are  fixed  at 
definite  points  in  the  side  of  a  pipe, 
as  represented  in  the  figure.  When  the 
pipe  gives  its  second  tone,  the  central 
flame  is  at  an  antinode  and  remains  un- 
affected, while  the  other  two,  being  at 
nodes,  are  agitated  or  blown  out.  When 
it  gives  its  first  tone,  the  central  flame, 
which  is  now  at  a  node,  is  more  power- 
fully affected  than  the  others.  The  gas 
which  supplies  these  burners  is  separated 
from  the  air  in  the  pipe  only  by  a  thin 
membrane.  When  the  pipe  is  made  to 
speak,  the  flame  at  the  node  is  violently 
agitated,  in  consequence  of  the  changes 
of  pressure  on  the  back  of  the  membrane, 
while  those  at  the  ventral  points  are 
scarcely  affected.  The  agitation  of  the 
flame  is  a  true  vibration;  and,  when  ex- 
amined by  the  aid  of  a  revolving  mirror, 
presents  the  appearance  of  tongues  of 

flame  alternating  with  nearly  dark  spaces.  If  two  pipes,  one  an 
octave  higher  than  the  other,  are  connected  with  the  same  gas  flame, 
or  with  two  gas  flames  which  can  be  viewed  in  the  same  mirror,  the 
tongues  of  flame  corresponding  to  the  upper  octave  are  seen  to  be 
twice  as  numerous  as  the  others. 


Fig.  631.— Manometric  Flames. 


CHAPTER    LXV. 


ANALYSIS   OF  VIBRATIONS.      CONSTITUTION   OF   SOUNDS. 


921.  Optical  Examination   of  Sonorous   Vibrations. — Sound  is  a 
special  sensation  belonging  to  the  sense  of  hearing;  but  the  vibra- 
tions which  are  its  physical  cause  often  manifest  themselves  to 
other  senses.     For  instance,  we  can  often  feel  the  tremors  of  a  sono- 
rous body  by  touching  it;  we  see  the  movements  of  the  sand  on  a 
vibrating  plate,  the  curve  traced  by  the  style  of  a  vibroscope,  &c. 
The  aid  which  one  sense  can  thus  furnish  in  what  seems  the  peculiar 
province   of   another   is  extremely  interesting.     M.  Lissajous   has 
devised  a  very  beautiful  optical   method  of   examining  sonorous 
vibrations,  which  we  will  briefly  describe. 

922.  Lissajous'  Experiment. — Suppose  we  introduce  into  a  dark 
room  (Fig.  632)  a  beam  of  solar  rays,  which,  after  passing  through 
a  lens  L,  is  reflected,  first,  from  a  small  mirror  fixed  on  one  of  the 
branches  of  a  tuning-fork  D,  and  then  from  a  second  mirror  M,  which 
throws  it  on  a  screen  E;  we  can  thus,  by  proper  adjustments,  form 
upon  the  screen  a  sharp  and  bright  image  of  the  sun,  which  will 
appear  as  a  small  spot  of  light.     As  long  as  the  apparatus  remains 
at  rest,  we  shall  not  observe  any  movement  of  the  image;  but  if  the 
tuning-fork   vibrates,  the  image  will   move  rapidly  up  and  down 
along  the  line  I,  I',  producing,  in  consequence  of  the  persistence  of 
impressions,  the  appearance  of  a  vertical  line  of  light.    If  the  tuning- 
fork  remains  at  rest,  but  the  mirror  M  is  rotated  through  a  small 
angle  about  a  vertical  axis,  the  image  will  move  horizontally.    Con- 
sequently, if  both  these  motions  take  place  simultaneousely,  the  spot 
of  light  will  trace  out  on  the  screen  a  sinuous  line,  as  represented  in 
the  figure,  each  S-shaped  portion  corresponding  to  one  vibration  of 
the  tuning-fork. 

Now,  let  the  mirror  M  be  replaced  by  a  small  mirror  attached  to 


928  ANALYSIS   OF   VIBRATIONS. 

a  second  tuning-fork,  which  vibrates  in  a  horizontal  plane,  as  in 


Fig.  632. — Principle  of  Lissajous'  Experiment. 

Fig.  633.     If  this  fork  vibrates  alone,  the  image  will  move  to  and 


Fig.  633. -Lissajous'  Experiment. 

fro  horizontally,  presenting  the  appearance  of  a  horizontal  line  of 


LISSAJOUS'   FIGUKES. 


929 


light,  which  gradually  shortens  as  the  vibrations  die  away.  If  both 
forks  vibrate  simultaneously,  the  spot  of  light  will  rise  and  fall  ac- 
cording to  the  movements  of  the  first  fork,  and  will  travel  left  and 
right  according  to  the  movements  of  the  second  fork.  The  curve 
actually  described,  as  the  resultant  of  these  two  component  motions, 
is  often  extremely  beautiful.  Some  varieties  of  it  are  represented  in 
Fig.  634. 

Instead  of  throwing  the  curves  on  a  screen,  we  may  see  them  by 
looking  into  the  second  mirror,  either  with  a  telescope,  as  in  Fig.  633, 


Fig.  634.-Lissajous'  Figures,  Uniion,  Octave,  and  Fifth. 

or  with  the  naked  eye.  In  this  form  of  the  experiment,  a  lamp  sur- 
rounded by  an  opaque  cylinder,  pierced  with  a  small  hole  just  opposite 
the  flame,  as  represented  in  the  figure,  is  a  very  convenient  source 
of  light. 

The  movement  of  the  image  depends  almost  entirely  on  the  angular 
movements  of  the  mirrors,  not  on  their  movements  of  translation; 
but  the  distinction  is  of  no  importance,  for,  in  the  case  of  such  small 
movements,  the  linear  and  angular  changes  may  be  regarded  as 
strictly  proportional. 

Either  fork  vibrating  alone  would  cause  the  image  to  execute 
simple  harmonic  motion  (§§  109-111),  or,  as  it  may  conveniently 


930  ANALYSIS   OF  VIBRATIONS. 

be  called,  simple  vibration;  so  that  the  movement  actually  executed 
will  be  the  resultant  of  two  simple  harmonic  motions  in  directions 
perpendicular  to  each  other. 

Suppose  the  two  forks  to  be  in  unison.  Then  the  two  simple  har- 
monic motions  will  have  the  same  period,  and  the  path  described 
will  always  be  some  kind  of  ellipse,1  the  circle  and  straight  line  being 
included  as  particular  cases.  It  will  be  a  straight  line  if  both  forks 
pass  through  their  positions  of  equilibrium  at  the  same  instant.  In 
order  that  it  may  be  a  circle,  the  amplitudes  of  the  two  simple  har- 
monic motions  must  be  equal,  and  one  fork  must  be  in  a  position  of 
maximum  displacement  when  the  other  is  in  the  position  of  equi- 
librium. 

If  the  unison  were  rigorous,  the  curve  once  obtained  would  remain 
unchanged,  except  in  so  far  as  its  breadth  and  height  became  reduced 
by  the  dying  away  of  the  vibrations.  But  this  perfect  unison  is 
never  attained  in  practice,  and  the  eye  detects  changes  depending 
on  differences  of  pitch  too  minute  to  be  perceived  by  the  ear.  These 
changes  are  illustrated  by  the  upper  row  of  forms  in  Fig.  634,  com- 
mencing, say,  with  the  sloping  straight  line  at  the  left  hand,  which 
gradually  opens  out  into  an  ellipse,  and  afterwards  contracts  into  a 
straight  line,  sloping  the  opposite  way.  It  then  retraces  its  steps, 
the  motion  being  now  in  opposition  to  the  arrows  in  the  figure,  and 
then  repeats  the  same  changes. 

If  the  interval  between  the  two  forks  is  an  octave,  we  shall  obtain 
the  curves  represented  in  the  second  row;2  if  the  interval  is  a  fifth, 
we  shall  obtain  the  curves  in  the  lowest  row.  In  each  case  the  order 
of  the  changes  will  be  understood  by  proceeding  from  left  to  right, 

1  Employing  horizontal  and  vertical  co-ordinates,  and  denoting  the  amplitudes  by  a  and 

b,  we  have,  in  the  case  of  unison,  -  =  sin  0,   ?  =  sin  (0  +  /3),  where  j3  denotes  the  difference 
a  b 

of  phase,  and  6  is  an  angle  varying  directly  as  the  time.  Eliminating  6,  we  obtain  the 
equation  to  an  ellipse,  whose  form  and  dimensions  depend  upon  the  given  quanti- 
ties, a,  b,  /3. 

a  The  middle  curve  in  this  row  is  a  parabola,  and  corresponds  to  the  elimination  of  0 

between  the  equations  ^=cos  20,  —=  cos  0.  The  coefficient  2  indicates  the  double 
frequency  of  horizontal  as  compared  with  vertical  vibrations. 

The  general  equations  to  Lissajous'  figures  are    -  =  sin  m  0,    -r=  sin  (n  0  +  p),  where  m 

and  n  are  proportional  to  the  frequencies  of  horizontal  and  vertical  vibrations.  The  gradual 
changes  from  one  figure  to  another  depend  on  the  gradual  change  of  ft  and  all  the  figures 
can  be  inscribed  in  a  rectangle,  whose  length  and  breadth  are  2  a  and  2  b. 


OPTICAL  TUNING.  931 

and  then  back  again;  but  the  curves  obtained  in  returning  will  be 
inverted. 

923.  Optical  Tuning. — By  the  aid  of  these  principles,  tuning-forks 
can  be  compared  with  a  standard  fork  with  much  greater  precision 
than  would  be  attainable  by  ear.  Fig.  635  represents  a  convenient 


Fig.  635. — Optical  Comparison  of  Tuning-forks. 

arrangement  for  this  purpose.  A  lens  /  is  attached  to  one  of  the 
prongs  of  a  standard  fork,  which  vibrates  in  a  horizontal  plane;  and 
above  it  is  fixed  an  eye-piece  g,  the  combination  of  the  two  being 
equivalent  to  a  microscope.  The  fork  to  be  compared  is  placed  up- 
right beneath,  and  vibrates  in  a  vertical  plane,  the  end  of  one  prong 
being  in  the  focus  of  the  microscope.  A  bright  point  m,  produced 
by  making  a  little  scratch  on  the  end  of  the  prong  with  a  diamond, 
is  observed  through  the  microscope,  and  is  illuminated,  if  necessary, 
by  converging  a  beam  of  light  upon  it  through  the  lens  c.  When 
the  forks  are  set  vibrating,  the  bright  point  is  seen  as  a  luminous 
ellipse,  whose  permanence  of  form  is  a  .test  of  the  closeness  of  the 
unison.  The  ellipse  will  go  through  a  complete  cycle  of  changes  in  the 
time  required  for  one  fork  to  gain  a  complete  vibration  on  the  other. 


932 


ANALYSIS  OF  VIBRATIONS. 


924.  Other  Modes  of  producing  Lissajous'  Figures. — An  arrange- 
ment devised  in  1844  by  Professor  Blackburn,  of  Glasgow,  then  a 
student  at  Cambridge,  affords  a  very  easy  mode  of  obtaining,  by  a 
slow  motion,  the  same  series  of  curves  which,  in  the  above  arrange- 
ments, are  obtained  by  a  motion  too  quick  for  the  eye  to  follow.  A 
cord  ABC  (Fig.  636)  is  fastened  at  A  and  C,  leaving  more  or  less 


»D 

Fig.  636. — Blackburn's  Pendulum. 

slack,  according  to  the  curves  which  it  is  desired  to  obtain;  and 
to  any  intermediate  point  B  of  the  cord  another  string  is  tied, 
carrying  at  its  lower  end  a  heavy  body  1)  to  serve  as  pendulum- 
bob. 

If,  when  the  system  is  in  equilibrium,  the  bob  is  drawn  aside  in 
the  plane  of  ABC  and  let  go,  it  will  execute  vibrations  in  that 
plane,  the  point  B  remaining  stationary,  so  that  the  length  of  the 
pendulum  is  BD.  If,  on  the  other  hand,  it  be  drawn  aside  in  a 
plane  perpendicular  to  the  plane  ABC,  it  will  vibrate  in  this  per- 
pendicular plane,  carrying  the  whole  of  the  string  with  it  in  its 
motion,  so  that  the  length  of  the  pendulum  is  the  distance  of  the 
bob  from  the  point  E,  in  which  the  straight  line  A  C  is  cut  by  D  B 
produced.  The  frequencies  of  vibration  in  the  two  cases  will  be 
inversely  as  the  square  roots  of  the  pendulum-lengths  B  D,  ED. 

If  the  bob  is  drawn  aside  in  any  other  direction,  it  will  not  vibrate 
in  one  plane,  but  will  perform  movements  compounded  of  the  two 
independent  modes  of  vibration  just  described,  and  will  thus  describe 
curves  identical  with  Lissajous'.  If  the  ratio  of  E  D  to  B  D  is  nearly 
equal  to  unity,  as  in  the  left-hand  figure,  we  shall  have  curves  cor- 
responding to  approximate  unison.  If  it  be  approximately  4,  as  in 
the  right-hand  figure,  we  shall  obtain  the  curves  of  the  octave. 
Traces  of  the  curves  can  be  obtained  by  employing  for  the  bob  a 


CHARACTER  OR  TIMBRE.  933 

vessel  containing  sand,  which  runs  out  through  a  funnel-shaped 
opening  at  the  bottom.1 

The  curves  can  also  be  exhibited  by  fixing  a  straight  elastic  rod 
at  one  end,  and  causing  the  other  end  to  vibrate  transversely.  This 
was  the  earliest  known  method  of  obtaining  them.  If  the  flexural 
rigidity  of  the  rod  is  precisely  the  same  for  all  transverse  directions, 
the  vibrations  will  be  executed  in  one  plane;  but  if  there  be  any 
inequality  in  this  respect,  there  will  be  two  mutually  perpendicular 
directions  possessing  the  same  properties  as  the  two  principal  direc- 
tions of  vibration  in  Blackburn's  pendulum.  A  small  bright  metal 
knob  is  usually  fixed  on  the  vibrating  extremity  to  render  its  path 
visible.  The  instrument  constructed  for  this  mode  of  exhibiting  the 
figures  is  called  a  kaleidophone.  In  its  best  form  (devised  by 
Professor  Barrett)  the  upper  and  lower  halves  of  the  rod  (which  is 
vertical)  are  flat  pieces  of  steel,  with  their  planes  at  right  angles, 
and  a  stand  is  provided  for  clamping  the  lower  piece  at  any  point 
of  its  length  that  may  be  desired,  so  as  to  obtain  any  required  com- 
bination. 

925.  Character. — Character  or  timbre,  which  we  have  already 
defined  in  §  889,  must  of  necessity  depend  on  the  form  of  the  vibra- 
tion of  the  aerial  particles  by  which  sound  is  transmitted,  the  word 
form  being  used  in  the  metaphorical  sense  there  explained,  for  in 
the  literal  sense  the  form  is  always  a  straight  line.  When  the 
changes  of  density  are  represented  by  ordinates  of  a  curve,  as  in 
Fig.  603,  the  form  of  this  curve  is  what  is  meant  by  the  form  of 
vibration. 

The  subject  of  timbre  has  been  very  thoroughly  investigated  in 
recent  years  by  Helmholtz;  and  the  results  at  which  he  has  arrived 
are  now  generally  accepted  as  correct. 

The  first  essential  of  a  musical  note  is,  that  the  aerial  movements 
which  constitute  it  shall  be  strictly  periodic;  that  is  to  say,  that 
each  vibration  shall  be  exactly  like  its  successor,  or  at  all  events, 
that,  if  there  be  any  deviation  from  strict  periodicity,  it  shall  be  so 
gradual  as  not  to  produce  sensible  dissimilarity  between  several  con- 
secutive vibrations  of  the  same  particle. 

There  is  scarcely  any  proposition  more  important  in  its  application 

1  Mr.  Hubert  Airy  has  obtained  very  beautiful  traces  by  attaching  a  glass  pen  to  the 
bob  (see  Nature,  Aug.  17  and  Sept  7,  1871),  and  in  Tisley's  harmonograph  the  same 
result  is  obtained  by  means  of  two  pendulums,  one  of  which  moves  the  paper  and  the 
other  the  pen. 


934  ANALYSIS  OF  VIBRATIONS. 

to  modern  physical  investigations  than  the  following  mathematical 
theorem,  which  was  discovered  by  Fourier: — Any  periodic  vibra- 
tion executed  in  one  line  can  be  definitely  resolved  into  simple 
vibrations,  of  which  one  has  the  same  frequency  as  the  given  vibra- 
tion, and  the  others  have  frequencies  2,  3,  4,  5  ...  times  as  great, 
no  fractional  multiples  being  admissible.  The  theorem  may  be 
briefly  expressed  by  saying  that  every  periodic  vibration  consists  of 
a  fundamental  simple  vibration  and  its  harmonics. 

We  cannot  but  associate  this  mathematical  law  with  the  experi- 
mental fact,  that  a  trained  ear  can  detect  the  presence  of  harmonics 
in  all  but  the  very  simplest  musical  notes.  The  analysis  which 
Fourier's  theorem  indicates,  appears  to  be  actually  performed  by  the 
auditory  apparatus. 

The  constitution  of  a  periodic  vibration  may  be  said  to  be  known 
if  we  know  the  ratios  of  the  amplitudes  of  the  simple  vibrations 
which  compose  it;  and  in  like  manner  the  constitution  of  a  sound 
may  be  said  to  be  known  if  we  know  the  relative  intensities  of  the 
different  elementary  tones  which  compose  it. 

Helmholtz  infers  from  his  experiments  that  the  character  of  a 
musical  note  depends  upon  its  constitution  as  thus  denned;  and 
that,  while  change  of  intensity  in  any  of  the  components  produces 
a  modification  of  character,  change  of  phase  has  no  influence  upon  it 
whatever.  Sir  W.  Thomson,  in  a  paper  "  On  Beats  of  Imperfect 
Harmonies,"1  adduces  strong  evidence  to  show  that  change  of  phase 
has,  in  some  cases  at  least,  an  influence  on  character. 

The  harmonics  which  are  present  in  a  note,  usually  find  their 
origin  in  the  vibrations  of  the  musical  instrument  itself.  In  the 
case  of  stringed  instruments,  for  example,  along  with  the  vibration 
of  the  string  as  a  whole,  a  number  of  segmental  vibrations  are  sim- 
ultaneously going  on.  Fig.  637  represents  curves  obtained  by  the 
composition  of  the  fundamental  mode  of  vibration  with  another  an 
octave  higher.  The  broken  lines  indicate  the  forms  which  the  string 
would  assume  if  yielding  only  its  fundamental  note.2  The  continu- 
ous lines  in  the  first  and  third  figures  are  forms  which  a  string  may 
assume  in  its  two  positions  of  greatest  displacement,  when  yielding 
the  octave  along  with  the  fundamental,  the  time  required  for  the 

1  Proc.  S.  S.  E.     1878. 

8  The  form  of  a  string  vibrating  so  as  to  give  only  one  tone  (whether  fundamental  or 
harmonic)  is  a  curve  of  sines,  all  its  ordinates  increasing  or  diminishing  in  the  same  pro- 
portion, as  the  string  moves. 


HARMONICS. 


935 


string  to  pass  from  one  of  these  positions  to  the  other  being  the  same 
as  the  time  in  which  each  of  its  two  segments  moves  across  and  back 


Hg.  637.— String  giving  first  Two  Tonea. 

again.  The  second  and  fourth  figures  must  in  like  manner  be  taken 
together,  as  representing  a  pair  of  extreme  positions.  The  number 
of  harmonics  thus  yielded  by  a  pianoforte  wire  is  usually  some  four 
or  five;  and  a  still  larger  number  are  yielded  by  the  strings  of  a  violin. 
The  notes  emitted  from  wide  organ-pipes  with  flute  mouth-pieces 
are  very  deficient  in  harmonics.  This  defect  is  remedied  by  combining 
with  each  of  the  larger  pipes  a  series  of  smaller  pipes,1  each  yielding 
one  of  its  harmonics.  An  ordinary  listener  hears  only  one  note,  of 
the  same  pitch  as  the  fundamental,  but  much  richer  in  character 
than  that  which  the  fundamental  pipe  yields  alone.  A  trained  ear 
can  recognize  the  individual  harmonics  in  this  case  as  in  any  other. 

1  The  stops  called  open  diapason  and  stop  diapason  (consisting  respectively  of  open  and 
stopped  pipes),  give  the  fundamental  tone,  almost  free  from  harmonics.  The  stop  absurdly 
called  principal  gives  the  second  tone,  that  is  the  octave  above  the  fundamental.  The 
stops  called  twelfth  and  fifteenth  give  the  third  and  fourth  tones,  which  are  a  twelfth 
(octave  +  fifth),  and  a  fifteenth  (double  octave)  above  the  fundamental.  The  fifth,  sixth, 
and  eighth  tones  are  combined  to  form  the  stop  called  mixture. 

As  many  of  our  readers  will  be  unacquainted  with  the  structure  of  organs,  it  may  be 
desirable  to  state  that  an  organ  contains  a  number  of  complete  instruments,  each  consisting 
of  several  octaves  of  pipes.  Each  of  these  complete  instruments  is  called  a  stop,  and  is 
brought  into  use  at  the  pleasure  of  the  organist  by  pulling  out  a  slide,  by  means  of  a  knob- 
handle,  on  which  the  name  of  the  stop  is  marked.  To  throw  it  out  of  use,  he  pushes  in 
the  slide.  A  large  number  of  stops  are  often  in  use  at  once. 


936  ANALYSIS  OF  VIBRATIONS. 

It  is  important  to  remark,  that  though  the  presence  of  harmonic 
subdivisions  in  a  vibrating  body  necessarily  produces  harmonics  in 
the  sound  emitted,  the  converse  cannot  be  asserted.  Simple  vibra- 
tions, executed  by  a  vibrating  body,  produce  vibrations  of  the  same 
frequency  as  their  own,  in  any  medium  to  which  they  are  trans- 
mitted, but  not  necessarily  simple  vibrations.  If  they  produce  com- 
pound vibrations,  these,  as  we  have  seen  (§  925),  must  consist  of  a 
fundamental  simple  vibration  and  its  harmonics. 

926.  Helmholtz's  Resonators. — Helmholtz  derived  material  aid  in 
his  researches  from  an  instrument  devised  by  himself,  and  called  a 
resonator  or  resonance  globe  (Fig.  638).  It  is  a  hollow  globe  of  thin 


Fig.  638.— Resonator. 

brass,  with  an  opening  at  each  end,  the  larger  one  serving  for  the 
admission  of  sound,  while  the  smaller  one  is  introduced  into  the  ear. 
The  inclosed  mass  of  air  has,  like  the  column  of  air  in  an  organ-pipe, 
a  particular  fundamental  note  of  its  own,  depending  upon  its  size; 
and  whenever  a  note  of  this  particular  pitch  is  sounded  in  its  neigh- 
bourhood, the  inclosed  air  takes  it  up  and  intensifies  it  by  resonance. 
In  order  to  test  the  presence  or  absence  of  a  particular  harmonic  in 
a  given  musical  tone,  a  resonator,  in  unison  with  this  harmonic,  is 
applied  to  the  ear,  and  if  the  resonator  speaks  it  is  known  that  the 
harmonic  is  present.  These  instruments  are  commonly  constructed 
so  as  to  form  a  series,  whose  notes  correspond  to  the  bass  C  of  a 
man's  voice,  and  its  successive  harmonics  as  far  as  the  10th  or  12th. 
Koenig  has  applied  the  principle  of  manometric  flames  to  enable  a 
large  number  of  persons  to  witness  the  analysis  of  sounds  by  resona- 
tors. A  series  of  6  resonators,  whose  notes  have  frequencies  propor- 
tional to  1,  2,  3,  4,  5,  6,  are  fixed  on  a  stand  (Fig.  639),  and  their 
smaller  ends,  instead  of  being  applied  to  the  ear,  are  connected  each 


CONSTITUTION. 


937 


with  a  separate  manometric  capsule,  which  acts  on  a  gas  jet.  When 
the  mirrors  are  turned,  it  is  easy  to  see  which  of  the  flames  vibrate 
while  a  sonorous  body  is  passed  in  front  of  the  resonatora 


Fig.  639. — Analysis  by  Manometric  Flames. 

A  simple  tone,  unaccompanied  by  harmonics,  is  dull  and  uninter- 
esting, and,  if  of  low  pitch,  is  very  destitute  of  penetrating  quality. 

Sounds  composed  of  the  first  six  elementary  tones  in  fair  propor- 
tion, are  rich  and  sweet. 

The  higher  harmonics,  if  sufficiently  subdued,  may  also  be  present 
without  sensible  detriment  to  sweetness,  and  are  useful  as  contribut- 
ing to  expression.  When  too  loud,  they  render  a  sound  harsh  and 
grating;  an  effect  which  is  easily  explained  by  the  discordant  com- 
binations which  they  form  one  with  another;  the  8th  and  9th  tones, 
for  example,  are  at  the  same  interval  as  the  notes  Do  and  Re. 

927.  Vowel  Sounds. — The  human  voice  is  extremely  rich  in  har- 
monics, as  may  be  proved  by  applying  the  series  of  resonators  to  the 


938  ANALYSIS  OF  VIBRATIONS. 

ear  while  the  fundamental  note  is  sung.  The  origin  of  the  tones  of 
the  voice  is  in  the  vocal  chords,  which,  when  in  use,  form  a  dia- 
phragm with  a  slit  along  its  middle.  The  edges  of  this  slit  vibrate 
when  air  is  forced  through,  and,  by  alternately  opening  and  closing 
the  passage,  perform  the  part  of  a  reed.  The  cavity  of  the  mouth 
serves  as  a  resonance  chamber,  and  reinforces  particular  notes  de- 
pending on  the  position  of  the  organs  of  speech.  It  is  by  this  reson- 
ance that  the  various  vowel  sounds  are  produced.  The  deepest  pitch 
belongs  to  the  vowel  sound  which  is  expressed  in  English  by  oo  (as 
in  moon),  and  the  highest  to  ee  (as  in  screech). 

Willis  in  18281  succeeded  in  producing  the  principal  vowel  sounds 
by  a  single  reed  fitted  to  various  lengths  of  tube.  Wheatstone,  a 
few  years  later,  made  some  advances  in  theory,2  and  constructed  a 
machine  by  which  nearly  all  articulate  sounds  could  be  imitated. 

Excellent  imitations  of  some  of  the  vowel  sounds  can  be  obtained 
by  placing  Helmholtz's  resonators,  one  at  a  time,  on  a  free-reed  pipe, 
the  small  end  of  the  resonator  being  inserted  in  the  hole  at  the  top 
of  the  pipe. 

The  best  determinations  of  the  particular  notes  which  are  rein- 
forced in  the  case  of  the  several  vowel  sounds,  have  been  made  by 
Helmholtz,  who  employed  several  methods,  but  chiefly  the  two  fol- 
lowing:— 

1.  Holding  resonators  to  the  ear,  while  a  particular  vowel  sound 
was  loudly  sung. 

2.  Holding  vibrating  tuning-forks  in  front  of  the  mouth  when  in 
the  proper  position  for  pronouncing  a  given  vowel;  and  observing 
which  of  them  had  their  sounds  reinforced  by  resonance.3 

Helmholtz  has  verified  his  determinations  synthetically.  He  em- 
ploys a  set  of  tuning-forks  which  are  kept  in  vibration  by  the  alter- 
nate making  and  unmaking  of  electro-magnets,  the  circuit  being 
made  and  broken  by  the  vibrations  of  one  large  fork  of  64  vibrations 
per  second.  The  notes  of  the  other  forks  are  the  successive  har- 
monics of  this  fundamental  note.  Each  fork  is  accompanied  by  a 

Cambridge  Transactions,  vol.  iiL 

2  London  and  Westminster  Review,  October,  1837. 

8  According  to  Koenig  (Comptes  Rendus,  1870)  the  notes  of  strongest  resonance  for  the 
vowels  u,  o,  a,  e,  i,  as  pronounced  in  North  Germany,  are  the  five  successive  octaves  of 
B  flat,  commencing  with  that  which  corresponds  to  the  space  above  the  top  line  of  the 
base  clef.  Willis,  Helmholtz,  and  Koenig  all  agree  as  regards  the  note  of  the  vowel  o, 
which  is  very  nearly  that  of  a  common  A  tuning-fork.  They  are  also  agreed  respesting 
the  note  of  a  (as  in  father),  which  is  an  octave  higher. 


PHONOGRAPH. 


939 


resonance-tube,  which,  when  open,  renders  the  note  of  the  fork 
audible  at  a  distance;  and  by  means  of  a  set  of  keys,  like  those  of  a 
piano,  any  of  these  tubes  can  be  opened  at  pleasure.  The  different 
vowel-sounds  can  thus  be  produced  by  employing  the  proper  com- 
binations. 

The  same  apparatus  served  for  establishing  the  principle  (§  925), 
that  the  character  of  a  musical  sound  depends  only  on  constitution, 
irrespective  of  change  of  phase. 

928.  Phonograph. — Mr.  Edison  of  New  York  has  been  successful 
in  constructing  an  instrument  which  can  reproduce  articulate  sounds 
spoken  into  it.  The  voice  of  the  speaker  is  directed  into  a  funnel, 
which  converges  the  sonorous  waves  upon  a  diaphragm  carrying  a 
style.  The  vibrations  of  the  diaphragm  are  impressed  by  means  of 
this  style  upon  a  sheet  of  tin-foil,  which  is  fixed  on  the  outside  of  a 
cylinder  to  which  a  spiral  motion  is  given  as  in  the  vibroscope  (Fig. 
616).  After  this  has  been  done,  the  cylinder  with  the  tin-foil  on  it  is 
shifted  back  to  its  original  position,  the  style  is  brought  into  contact 


Fig.  640. -Phonograph. 

with  the  tin-foil  as  at  first,  and  the  cylinder  is  then  turned  as  before. 
The  indented  record  is  thus  passed  beneath  the  style,  and  forces  it 
and  the  attached  diaphragm  to  execute  movements  resembling  their 
original  movements.  The  diaphragm  accordingly  emits  sounds  which 
are  imitations  of  those  previously  spoken  to  it.  Tunes  sung  into  the 
funnel  are  thus  reproduced  with  great  fidelity,  and  sentences  clearly 
spoken  into  it  are  reproduced  with  sufficient  distinctness  to  be 
understood. 

The  instrument  is  represented  in  Fig.  640.   By  turning  the  handle 
E,  which  is  attached  to  a  massive  fly-wheel,  the  cylinder  B  is  made 


940  ANALYSIS   OF  VIBRATIONS. 

to  revolve  and  at  the  same  time  to  travel  longitudinally,  as  the  axle 
on  which  it  is  mounted  is  a  screw  working  in  a  fixed  nut.  The  sur- 
face of  the  cylinder  is  also  fluted  screw-fashion,  the  distance  between 
its  flutings  being  the  same  as  the  distance  between  the  threads  on 
the  axle.  A  is  the  diaphragm,  of  thin  sheet-iron,  having  the  style 
fixed  to  its  centre  but  not  visible  in  the  figure.  The  diaphragm  and 
funnel  are  carried  by  the  frame  D,  which  turns  on  a  hinge  at  the 
bottom.  C  C  are  adjusting  screws  for  bringing  the  style  exactly 
opposite  the  centre  of  the  groove  on  the  cylinder,  and  another  screw 
is  provided  beneath  the  frame  D,  for  making  the  style  project  so 
far  as  to  indent  the  tin-foil  without  piercing  it.  The  tin-foil  is  put 
round  the  cylinder,  and  lightly  fastened  with  cement,  so  that  it  can 
be  quickly  taken  off  and  changed. 

In  another  form  of  the  instrument,  the  rotation  of  the  cylinder  is 
effected  by  means  of  a  driving  weight  and  governor,  which  give  it  a 
constant  velocity.  This  is  a  great  advantage  in  reproducing  music, 
but  is  of  little  or  no  benefit  for  speech. 


CHAPTER    LXVI. 


CONSONANCE,  DISSONANCE,  AND  RESULTANT  TONES. 


929.  Concord  and  Discord.— Every  one  not  utterly  destitute  of 
musical  ear  is  familiar  with  the  fact  that  certain  notes,  when  sounded 
together,  produce  a  pleasing  effect  by  their  combination,  while  cer- 
tain others  produce  an  unpleasing  effect.  The  combination  of  two 
or  more  notes,  when  agreeable,  is  called  concord  or  consonance; 
when  disagreeable,  discord  or  dissonance.  The  distinction  is  found 
to  depend  almost  entirely  on  difference  of  pitch,  that  is,  on  relative 
frequency  of  vibration;  so  that  the  epithets  consonant  and  dissonant 
can  with  propriety  be  applied  to  intervals. 

The  following  intervals  are  consonant:  unison  (1:1),  octave  (1 :  2), 
octave  +  fifth  (1 :  3),  double  octave  (1 :  4),  fifth  (2  :  3),  fourth  (3  :  4). 

The  major  third  (4  :  5)  and  major  sixth  (3 :  5),  together  with  the 
minor  third  (5  :  6)  and  minor  sixth  (5  :  8),  are  less  perfect  in  their 
consonance. 

The  second  and  the  seventh,  whether  major  or  minor,  are  dis- 
sonant intervals,  whatever  system  of  temperament  be  employed,  as 
are  also  an  indefinite  number  of  other  intervals  not  recognized  in 
music. 

Besides  the  difference  as  regards  pleasing  or  unpleasing  effect,  it 
is  to  be  remarked  that  consonant  intervals  can  be  identified  by  the  ear 
with  much  greater  accuracy  than  those  which  are  dissonant.  Musi- 
cal instruments  are  generally  tuned  by  octaves  and  fifths,  because 
very  slight  errors  of  excess  or  defect  in  these  intervals  are  easily 
detected  by  the  ear.  To  tune  a  piano  by  the  mere  comparison 
of  successive  notes  would  be  beyond  the  power  of  the  most  skilful 
musician.  A  sharply  marked  interval  is  always  a  consonant 
interval. 


942  CONSONANCE,   DISSONANCE,   AND   RESULTANT  TONES. 

930.  Jarring  Effect  of  Dissonance. — According  to  the  theory  pro- 
pounded by  Helmholtz,  the  unpleasant  effect  of  a  dissonant  interval 
consists  essentially  in  the  production  of  beats.    These  have  a  jarring 
effect  upon  the  auditory  apparatus,  which  becomes  increasingly  dis- 
agreeable as  the  beats  increase  in  frequency  up  to  a  certain  limit 
(about  33  per  second   for  notes  of   medium   pitch),  and  becomes 
gradually  less  disagreeable  as  the  frequency  is  still  further  increased. 
The  sensation  produced  by  beats  is  comparable  to  that  which  the 
eye  experiences  from  the  bobbing  of  a  gas  flame  in  a  room  lighted 
by  it;  but  the  frequency  which  entails  the  maximum  of  annoyance 
is  smaller  for  the  eye  than  for  the  ear,  on  account  of  the  greater  per- 
sistence of  visible  impressions.    The  annoyance  must  evidently  cease 
when  the  succession  becomes  so  rapid  as  to  produce  the  effect  of  a 
continuous  impression. 

We  have  already  (§  888)  described  a  mode  of  producing  beats  with 
any  degree  of  frequency  at  pleasure;  and  this  experiment  is  one  of 
the  main  foundations  on  which  Helmholtz  bases  his  view. 

931.  Beats  of  Harmonics. — The   beats  in  the  experiment  above 
alluded  to,  are  produced  by  the  imperfect  unison  of  two  notes,  and 
indicate  the  number  of  vibrations  gained  by  one  note  upon  the  other. 
Their  existence  is  easily  and  completely  explained  by  the  considera- 
tions adduced  in  §  888.     But  it  is  well  known  to  musicians,  and 
easily  established  by  experiment,  that  beats  are  also  produced  be- 
tween notes  whose  interval  is  approximately  an  octave,  a  fifth,  or 
some  other  consonance;  and  that,  in  these  cases  also,  the  beats  become 
more  rapid  as  the  interval  becomes  more  faulty. 

These  beats  are  ascribed  by  Helmholtz  to  the  common  harmonic 
of  the  two  fundamental  notes.  For  example,  in  the  case  of  the  fifth 
(2  :  3),  the  third  tone  of  the  lower  note  would  be  identical  with  the 
second  tone  of  the  upper,  if  the  interval  were  exact;  and  the  beats 
which  occur  are  due  to  the  imperfect  unison  consequent  on  the  devia- 
tion from  exact  truth.  All  beats  are  thus  explained  as  due  to  im- 
perfect unison. 

This  explanation  is  not  merely  conjectural,  but  is  established  by 
the  following  proofs: — 

1.  When  an  arrangement  is  employed  by  which  the  fifth  is  made 
false  by  a  known  amount,  the  number  of  beats  is  found  to  agree 
with  the  above  explanation.  Thus,  if  the  interval  is  made  to  cor- 
respond to  the  ratio  200  :  301,  it  is  observed  that  there  are  2  beats  to 
every  200  vibrations  of  the  lower  note.  Now  the  harmonics  which 


BEATS.  943 

are  in  approximate  unison  are  represented  by  600  and  602,  and  the 
difference  of  these  is  2. 

2.  When  the  resonator  corresponding  to  this  common  harmonic  is 
held  to  the  ear,  it  responds  to  the  beats,  showing  that  this  harmonic 
is  undergoing  variations  of  strength;  but  when  a  resonator  corre- 
sponding to  either  of  the  fundamental  notes  is  employed,  it  does  not 
respond  to  the  beats,  but  indicates  steady  continuance  of  its  appro- 
priate note. 

3.  By  a  careful  exercise  of  attention,  a  person  with  a  good  ear  can 
hear,  without  any  artificial  aids,  that  it  is  the  common  harmonic 
which  undergoes  variations  of  intensity,  and  that  the  fundamental 
notes  continue  steady. 

932.  Beating  Notes  must  be  Near  Together. — In  order  that  two 
simple  tones  may  yield  audible  beats,  it  is  necessary  that  the  musical 
interval  between  them  should  be  small;  in  other  words,  that  the  ratio 
of  their  frequencies  of  vibration  should  be  nearly  equal  to  unity. 
Two  simple  notes  of  300  and  320  vibrations  per  second  will  yield  20 
beats  in  a  second,  and  will  be  eminently  discordant,  the  interval 
between  them  being  only  a  semitone  (15  : 16),  but  simple  notes  of 
40  and  60  vibrations  per  second  will  not  give  beats,  the  interval 
between  them  being  a  fifth  (2  :  3).  The  wider  the  interval  between 
two  simple  notes,  the  feebler  will  be  their  beats;  and  accordingly, 
for  a  given  frequency  of  beats,  the  harshness  of  the  effect  increases 
with  the  nearness  of  the  notes  to  each  other  on  the  musical  scale.1 
By  taking  joint  account  of  the  number  of  beats  and  the  nearness  of 
the  beating  tones,  Helmholtz  has  endeavoured  to  express  numeri- 
cally the  severity  of  the  discords  resulting  from  the  combination  of 
the  note  C  of  256  vibrations  per  second  with  any  possible  note  lying 
within  an  octave  on  the  upper  side  of  it,  a  particular  constitution 
(approximately  that  of  the  violin)  being  assumed  for  both  notes. 
He  finds  a  complete  absence  of  discord  for  the  intervals  of  uni- 
son, the  octave,  and  the  fifth,  and  very  small  amounts  of  discord 
for  the  fourth,  the  sixth,  and  the  third.  By  far  the  worst  discords 
are  found  for  the  intervals  of  the  semitone  and  major  seventh, 


1  The  explanation  adopted  by  Helmholtz  is,  that  a  certain  part  of  the  ear — the  mem- 
brana  basilaris — is  composed  of  tightly  stretched  elastic  fibres,  each  of  which  is  attuned  to 
a  particular  simple  tone,  and  is  thrown  into  vibration  when  this  tone,  or  one  nearly  in 
unison  with  it,  is  sounded.  Two  tones  in  approximate  unison,  when  sounded  together, 
affect  several  fibres  in  common,  and  cause  them  to  beat.  Tones  not  in  approximate  unison 
affect  entirely  distinct  sets  of  fibres,  and  thus  cannot  produce  interference. 


944  CONSONANCE,  DISSONANCE,  AND  RESULTANT  TONES. 

and  the  next  worst  are  for  intervals  a  little  greater  or  less  than  the 
fifth. 

933.  Imperfect  Concord. — When  there  is  a  complete  absence  of 
discord  between  two  notes,  they  are  said  to  form  a  perfect  concord. 
The  intervals  unison,  fifth,  octave,  octave  +  fifth,  and  the  interval 
from  any  note  to  any  of  its  harmonics,  are  of  this  class.     The  third, 
fourth,  and  sixth  are  instances  of  imperfect  concord.     Suppose,  for 
example,  that  the  two  notes  sounded  together  are  C  of  256  and  E 
of  320  vibrations  per  second,  the  interval  between  these  notes  being 
a  true  major  third  (4:5);  and  suppose  each  of  these  notes  to  consist 
of  the  first  six  simple  tones. 

The  first  six  multiples  of  4  are 

4,  8,  12,  16,  20,  24. 
The  first  six  multiples  of  5  are 

5,  10,  15,  20,  25,  SO. 

In  searching  for  elements  of  discord,  we  select  (one  from  each  line) 
two  multiples  differing  by  unity. 

Those  which  satisfy  this  condition  are 

4  and  5;  16  and  15;  24  and  25. 

But  the  first  pair  (4  and  5)  may  be  neglected,  because  their  ratio 
differs  too  much  from  unity.  Discordance  will  result  from  each  of 
the  two  remaining  pairs;  that  is  to  say,  the  4th  element  of  the  lower 
of  our  two  given  notes  is  in  discordance  with  the  3d  element  of  the 
upper;  and  the  6th  element  of  the  lower  is  in  discordance  with  the 
5th  element  of  the  higher.  To  find  the  frequencies  of  the  beats,  we 
must  multiply  all  these  numbers  by  64,  since  256  is  4  times  64,  and 
320  is  5  times  64.  Instead  of  a  difference  of  1,  we  shall  then  find  a 
difference  of  64,  that  is  to  say,  the  number  of  beats  per  second  is  64 
in  the  case  of  each  of  the  two  discordant  combinations  which  we 
have  been  considering. 

934.  Resultant  Tones. — Under  certain  conditions  it  is  found  that 
two  notes,  when  sounded  together,  produce  by  their  combination 
other  notes,  which  are  not  constituents  of  either.     They  are  called 
resultant  tones,  and  are  of  two  kinds,  difference-tones  and  summa- 
tion-tones.    A  difference-tone  has  a  frequency  of  vibration  which 
is  the  difference  of  the  frequencies  of  its  components.     A  summa- 
tion-tone has  a  frequency  of  vibration  which  is  the  sum  of  the 


RESULTANT  TONES.  945 

frequencies  of  its  components.  As  the  components  may  either  be 
fundamental  tones  or  overtones,  two  notes  which  are  rich  in  har- 
monies may  yield,  by  their  combination,  a  large  number  of  resultant 
tones. 

The  difference-tones  were  observed  in  the  last  century  by  Sorge 
and  by  Tartini,  and  were,  until  recently,  attributed  to  beats.  The  fre- 
quency of  beats  is  always  the  difference  of  the  frequencies  of  vibra- 
tion of  the  two  elementary  tones  which  produce  them;  and  it  was 
supposed  that  a  rapid  succession  of  beats  produced  a  note  of  pitch 
corresponding  to  this  frequency. 

This  explanation,  if  admitted,  would  furnish  an  exception  to  what 
otherwise  appears  to  be  the  universal  law,  that  every  elementary 
tone  arises  from  a  corresponding  simple  vibration.1  Such  an  excep- 
tion should  not  be  admitted  without  necessity;  and  in  the  present 
instance  it  is  not  only  unnecessary,  but  also  insufficient,  inasmuch  as 
it  fails  to  render  any  account  of  the  summation-tones. 

Helmholtz  has  shown,  by  a  mathematical  investigation,  that  when 
two  systems  of  simple  waves  agitate  the  same  mass  of  air,  their 
mutual  influence  must,  according  to  the  recognized  laws  of  dynamics, 
give  rise  to  two  derived  systems,  having  frequencies  which  are  re- 
spectively the  sum  and  the  difference  of  the  frequencies  of  the  two 
primary  systems.  Both  classes  of  resultant  tones  are  thus  completely 
accounted  for. 

The  resultant  tones — especially  the  summation-tones,  which  are 
fainter  than  the  others — are  only  audible  when  the  primary  tones 
are  loud;  for  their  existence  depends  upon  small  quantities  of  the 
second  order,  the  amplitudes  of  the  primaries  being  regarded  (in 
comparison  with  the  wave-lengths)  as  small  quantities  of  the  first 
order. 

If  any  further  proof  be  required  that  the  difference  tones  are  not 
due  to  the  coalescence  of  beats,  it  is  furnished  by  the  fact  that,  under 
favourable  conditions,  the  rattle  of  the  beats  and  the  booming  of  the 
difference- tones  can  both  be  heard  together. 

935.  Beats  due  to  Resultant  Tones. — The  existence  of  resultant 
tones  serves  to  explain,  in  certain  cases,  the  production  of  beats 
between  notes  which  are  wanting  in  harmonics.  For  example,  if 
two  simple  sounds,  of  100  and  201  vibrations  per  second  respectively, 
are  sounded  together,  one  beat  per  second  will  be  produced  between 

1  The  discovery  of  this  law  is  due  to  Ohm. 
60 


946  CONSONANCE,   DISSONANCE,  AND  RESULTANT  TONES. 

the  difference-tone  of  101  vibrations  and  the  primary  tone  of  100 
vibrations.  By  the  beats  to  which  they  thus  give  rise,  resultant 
tones  exercise  an  influence  on  consonance  and  dissonance. 

Resultant  tones,  when  sufficiently  loud,  are  themselves  capable  of 
performing  the  part  of  primaries,  and  yielding  what  are  called  result- 
ant tones  of  the  second  order,  by  their  combination  with  other  pri- 
maries. Several  higher  orders  of  resultant  tones  can,  under  pecu- 
liarly favourable  circumstances,  be  sometimes  detected. 


OPTICS. 


CHAPTER   LXVII. 

PROPAGATION     OF    LIGHT. 

936.  Light. — Light  is  the  immediate  external  cause  of  our  visual 
impressions.  Objects,  except  such  as  are  styled  self-luminous, 
become  invisible  when  brought  into  a  dark  room.  The  presence  of 
something  additional  is  necessary  to  render  them  visible,  and  that 
mysterious  agent,  whatever  its  real  nature  may  be,  we  call  light. 

Light,  like  sound,  is  believed  to  consist  in  vibration;  but  it  does 
not,  like  sound,  require  the  presence  of  air  or  other  gross  matter  to 
enable  its  vibrations  to  be  propagated  from  the  source  to  the  per- 
cipient. When  we  exhaust  a  receiver,  objects  in  its  interior  do  not 
become  less  visible;  and  the  light  of  the  heavenly  bodies  is  not  pre- 
vented from  reaching  us  by  the  highly  vacuous  spaces  which  lie 
between. 

It  seems  necessary  to  assume  the  existence  of  a  medium  far  more 
subtle  than  ordinary  matter;  a  medium  which  pervades  alike  the 
most  vacuous  spaces  and  the  interior  of  all  bodies,  whether  solid, 
liquid,  or  gaseous;  and  which  is  so  highly  elastic,  in  proportion  to  its 
density,  that  it  is  capable  of  transmitting  vibrations  with  a  velocity 
enormously  transcending  that  of  sound. 

This  hypothetical  medium  is  called  aether.  From  the  extreme 
facility  with  which  bodies  move  about  in  it,  we  might  be  disposed 
to  call  it  a  subtle  fluid;  but  the  undulations  which  it  serves  to 
propagate  are  not  such  as  can  be  propagated  by  fluids.  Its  elastic 
properties  are  rather  those  of  a  solid;  and  its  waves  are  analogous 
to  the  pulses  which  travel  along  the  wires  of  a  piano  rather  than  to 
the  waves  of  extension  and  compression  by  which  sound  is  propa- 
gated through  air.  Luminous  vibrations  are  transverse,  while  those 
of  sound  are  longitudinal. 

A  self-luminous  body,  such  as  a  red-hot  poker  or  the  flame  of  a 


948  PROPAGATION   OF   LIGHT. 

candle,  is  in  a  peculiar  state  of  vibration.  This  vibration  is  com- 
municated to  the  surrounding  aether,  and  is  thus  propagated  to  the 
eye,  enabling  us  to  see  the  body.  In  the  majority  of  cases,  however, 
we  see  bodies  not  by  their  own  but  by  reflected  light;  and  we  are 
enabled  to  recognize  the  various  kinds  of  bodies  by  the  different 
modifications  which  light  undergoes  in  reflection  from  their  surfaces. 

As  all  bodies  can  become  sonorous,  so  also  all  bodies  can  become 
self-luminous.  To  render  them  so,  it  is  only  necessary  to  raise  them 
to  a  sufficiently  high  temperature,  whether  by  the  communication 
of  heat  from  a  furnace,  or  by  the  passage  of  an  electric  current,  or 
by  causing  them  to  enter  into  chemical  combination.  It  is  to  chemi- 
cal combination,  in  the  active  form  of  combustion,  that  we  are  in- 
debted for  all  the  sources  of  artificial  light  in  ordinary  use. 

The  vibrations  of  the  aether  are  capable  of  producing  other  effects 
besides  illumination.  They  constitute  what  is  called  radiant  heat, 
and  they  are  also  capable  of  producing  chemical  effects,  as  in  photo- 
graphy. Vibrations  of  high  frequency,  or  short  period,  are  the  most 
active  chemically.  Those  of  low  frequency  or  long  period  have 
usually  the  most  powerful  heating  effects;  while  those  which  affect 
the  eye  with  the  sense  of  light  are  of  moderate  frequency. 

937.  Rectilinear  Propagation  of  Light. — All  the  remarks  which 
have  'been  made  respecting  the  relations  between  period,  frequency, 
and  wave-length,  in  the  case  of  sound,  are  equally  applicable  to  light, 
inasmuch  as  all  kinds  of  luminous  waves  (like  all  kinds  of  sonorous 
waves)  have  sensibly  the  same  velocity  in  air;  but  this  velocity  is 
many  hundreds  of  thousands  of  times  greater  for  light  than  for  sound, 
and  the  wave-lengths  of  light  are  at  the  same  time  very  much  shorter 
than  those  of  sound.  Frequency,  being  the  quotient  of  velocity  by 
wave-length,  is  accordingly  about  a  million  of  millions  of  times 
greater  for  light  than  for  sound.  The  colour  of  lowest  pitch  is  deep 
red,  its  frequency  being  about  400  million  million  vibrations  per 
second,  and  its  wave-length  in  air  760  millionths  of  a  millimetre. 
The  colour  of  highest  pitch  is  deep  violet;  its  frequency  is  about  7GO 
million  million  vibrations  per  second,  and  its  wave-length  in  air  400 
millionths  of  a  millimetre.  It  thus  appears  that  the  range  of  seeing  is 
much  smaller  than  that  of  hearing,  being  only  about  one  octave. 

The  excessive  shortness  of  luminous  as  compared  with  sonorous 
waves  is  closely  connected  with  the  strength  of  the  shadows  cast  by 
a  light,  as  compared  with  the  very  moderate  loss  of  intensity  pro- 
duced by  interposing  an  obstacle  in  the  case  of  sound.  Sound  may, 


RECTILINEAR   PROPAGATION. 


949 


Fig.  641. -Rectilinear  Propagation. 


for  ordinary  purposes,  be  said  to  be  capable  of  turning  a  corner,  and 
light  to  be  only  capable  of  travelling  in  straight  lines.  The  latter 
fact  may  be  established  by  such  an  arrangement  as  is  represented  in 
Fig.  641.  Two  screens, 
each  pierced  with  a 
hole,  are  arranged  so 
that  these  holes  are  in 
a  line  with  the  flame 
of  a  candle.  An  eye 
placed  in  this  line,  be- 
hind the  screens,  is  then 
able  to  see  the  flame; 
but  a  slight  lateral  dis- 
placement, either  of  the 
eye,  the  candle,  or  either 
of  the  screens,  puts 
the  flame  out  of  sight. 
It  is  to  be  noted  that, 

in  this  experiment,  the  same  medium  (air)  extends  from  the  eye 
to  the  candle.  We  shall  hereafter  find  that,  when  light  has  to 
pass  from  one  medium  to  another,  it  is  often  bent  out  of  a  straight 
line. 

We  have  said  that  the  strength  of  light-shadows  as  compared  with 
sound-shadows  is  connected  with  the  shortness  of  luminous  waves. 
Theory  shows  that,  if  light  is  transmitted  through  a  hole  or  slit, 
whose  diameter  is  a  very  large  multiple  of  the  length  of  a  light- 
wave, a  strong  shadow  should  be  cast  in  all  oblique  directions;  but 
that,  if  the  hole  or  slit  is  so  narrow  that  its  diameter  is  comparable 
to  the  length  of  a  wave,  a  large  area  not  in  the  direct  path  of  the 
beam  will  be  illuminated.  The  experiment  is  easily  performed  in  a 
dark  room,  by  admitting  sunlight  through  an  exceedingly  fine  slit, 
and  receiving  it  on  a  screen  of  white  paper.  The  illuminated  area 
will  be  marked  with  coloured  bands,  called  diffraction-fringes;  and 
if  the  slit  is  made  narrower,  these  bands  become  wider. 

On  the  other  hand,  Colladon,  in  his  experiments  on  the  transmis- 
sion of  sound  through  the  water  of  the  Lake  of  Geneva,  established 
the  presence  of  a  very  sharply  defined  sound-shadow  in  the  water, 
behind  the  end  of  a  projecting  wall. 

For  the  present  we  shall  ignore  diffraction,1  and  confine  our  atten- 

1  See  Chap.  Ixxiv. 


950 


PROPAGATION   OF  LIGHT. 


tion  to  the  numerous  phenomena  which  result  from  the  rectilinear 
propagation  of  light. 

938.  Images  produced  by  Small  Apertures. — If  a  white  screen  is 
placed  opposite  a  hole  in  the  shutter  of  a  room  otherwise  quite  dark, 


Fig.  642.— Image  formed  by  Small  Aperture. 

an  inverted  picture  of  the  external  landscape  will  be  formed  upon  it, 
in  the  natural  colours.     The  outlines  will  be  sharper  in  proportion 
as  the  hole  is  smaller,  and  distant  objects  will 
be  more  distinctly  represented  than  those  which 
are  very  near. 

These  results  are  easily  explained.    Consider, 
>A  in  fact,  an  external  object  AB  (Fig.  643),  and 
let  O  be  the  hole  in  the  shutter.     The  point  A 
sends  rays  in  all  directions  into  space,  and  among 
them  a  small  pencil,  which,  after  passing  through 
the  opening  0,  falls  upon  the  screen  at  A'.     A' 
receives  light  from  no  other  point  but  A,  and 
A  sends  light  to  no  part  of  the  screen  except  A'. 
The  colour  and  brightness  of  the  spot  A'  will  accordingly  depend 
upon  the  colour  and  brightness  of  A;  in  other  words,  A'  will  be  the 


Fig.  643.— Explanation. 


IMAGES   FORMED   BY   SMALL  HOLES. 


951 


image  of  A.  In  like  manner  B'  will  be  the  image  of  B,  and  points  of 
the  object  between  A  and  B  will  have  their  images  between  A'  and  B'. 
An  inverted  image  A'B'  will  thus  be  formed  of  the  object  AB. 

As  the  image  thus  formed  of  an  external  point  is  not  a  point,  but 
a  spot,  whose  size  increases  with  that  of  the  opening,  there  must 
always  be  a  little  blurring  of  the  outlines  from  the  overlapping  of 
the  spots  which  represent  neighbouring  points  ;  but  this  will  be  com- 
paratively slight  if  the  opening  is  very  small. 

An  experiment,  substantially  the  same  as  the  above,  may  be  per- 
formed by  piercing  a  card  with  a  large  pin-hole,  and  holding  it  between 


Fig.  644. — Image  formed  by  Hole  in  a  Card. 

a  candle  and  a  screen,  as  in  Fig.  644.     An  inverted  image  of  the 
candle  will  thus  be  formed  upon  the  screen. 

When  the  sun  shines  through  a  small  hole  into  a  room  with  the 
blinds  down  (Fig.  645),  the  cone  of  rays  thus  admitted  is  easily 
traced  by  the  lighting  up  of  the  particles  of  dust  which  lie  in  its 
course.  The  image  of  the  sun  which  is  formed  at  its  further  ex- 
tremity will  be  either  circular  or  elliptical,  according  as  the  incidence 
of  the  rays  is  normal  or  oblique.  Fine  images  of  the  sun  are  some- 
times thus  formed  by  the  chinks  of  a  venetian-blind,  especially  when 
the  sun  is  low,  and  there  is  a  white  wall  opposite  to  receive  the 


052  PROPAGATION    OF   LIGHT. 

image.     In  these  circumstances  it  is  sometimes  possible  to  detect  the 
presence  of  spots  on  the  sun  by  examining  the  image. 

When  the  sun's  rays  shine  through  the  foliage  of  a  tree  (Fig.  646), 
the  spots  of  light  which  they  form  upon  the  ground  are  always  round 
or  oval,  whatever  may  be  the  shape  of  the  interstices  through  which 
they  have  passed,  provided  always  that  these  interstices  are  small. 
When  the  sun  is  undergoing  eclipse,  the  progress  of  the  eclipse  can 


Fig.  645. — Conical  Sunbeam. 

be  traced  by  watching  the  shape  of  these  images,  which  resembles 
that  of  the  uneclipsed  portion  of  the  sun's  disc. 

939.  Theory  of  Shadows. — The  rectilinear  propagation  of  light  is 
the  foundation  of  the  geometry  of  shadows.  Let  the  source  of  light 
be  a  luminous  point,  and  let  an  opaque  body  be  placed  so  as  to  inter- 
cept a  portion  of  its  rays  (Fig.  647).  If  we  construct  a  conical 
surface  touching  the  body  all  round,  and  having  its  vertex  at  the 
luminous  point,  it  is  evident  that  all  the  space  within  this  surface  on 
the  further  side  of  the  opaque  body  is  completely  screened  from  the 
rays.  The  cone  thus  constructed  is  called  the  shadow-cone,  and  its 
intersection  with  any  surface  behind  the  opaque  body  defines  the 
shadow  cast  upon  that  surface.  In  the  case  which  we  have  been 
supposing — that  of  a  luminous  point — the  shadow-cone  and  the 
shadow  itself  will  be  sharply  defined. 


SHADOWS. 


953 


Fig.  646.  —Images  of  Sun  formed  by  Foliage. 


954  PROPAGATION   OF  LIGHT. 

Actual  sources  of  light,  however,  are  not  mere  luminous  points, 
but  have  finite  dimensions.  Hence  some  complication  arises.  Con- 
sider, in  fact  (Fig.  648),  a  luminous  body  situated  between  two  opaque 
bodies,  one  of  them  larger,  and  the  other  smaller  than  itself.  Con- 
ceive a  cone  touching  the  luminous  body  and  either  of  the  opaque 
bodies  externally.  This  will  be  the  cone  of  total  shadow,  or  the 
cone  of  the  umbra.  All  points  lying  within  it  are  completely  ex- 
cluded from  view  of  the  luminous  body.  This  cone  narrows  or  en- 
larges as  it  recedes,  according  as  the  opaque  body  is  smaller  or  larger 
than  the  luminous  body.  In  the  former  case  it  terminates  at  a 
finite  distance.  In  the  latter  case  it  extends  to  infinite  distance. 

Now  conceive  a  double  cone  touching  the  luminous  body  and 
either  of  the  opaque  bodies  internally.  This  cone  will  be  wider 
than  the  cone  of  total  shadow,  and  will  include  it.  It  is  called  the 


Fig.  648. — Umbra  and  Penumbra. 

cone  of  partial  shadow,  or  the  cone  of  the  penumbra.  All  points 
lying  within  it  are  excluded  from  the  view  of  some  portion  of  the 
luminous  body,  and  are  thus  partially  shaded  by  the  opaque  body. 
If  they  are  near  its  outer  boundary,  they  are  very  slightly  shaded. 
If  they  are  so  far  within  it  as  to  be  near  the  total  shadow,  they  are 
almost  completely  shaded.  Accordingly,  if  the  shadow  of  the  opaque 
body  is  received  upon  a  screen,  it  will  not  have  sharply  defined 
edges,  but  will  show  a  gradual  transition  from  the  total  shadow 
which  covers  a  finite  central  area  to  a  complete  absence  of  shadow 
at  the  outer  boundary  of  the  penumbra.  Thus  neither  the  edges 
of  the  umbra  nor  those  of  the  penumbra  are  sharply  defined. 

The  umbra  and  penumbra  show  themselves  on  the  surface  of  the 


VELOCITY   OF   LIGHT.  955 

opaque  body  itself,  the  line  of  contact  of  the  umbral  cone  being 
farther  back  from  the  source  of  light  than  the  line  of  contact  of  the 
penumbral  cone.  The  zone  between  these  two  lines  is  in  partial 
shadow,  and  separates  the  portion  of  the  surface  which  is  in  total 
shadow  from  the  part  which  is  not  shaded  at  all. 

940.  Velocity  of  Light. — Luminous  undulations,  unlike  those  of 
sound,  advance  with  a  velocity  which  may  fairly  be  styled  incon- 
ceivable, being  about   300  million  metres   per  second,  or  186,000 
miles  per  second.     As  the  circumference  of  the  earth  is  only  40  mil- 
lion metres,  light  would  travel  seven  and  a  half  times  round  the 
earth  in  a  second. 

Hopeless  as  it  might  appear  to  attempt  the  measurement  of  such 
an  enormous  velocity  by  mere  terrestrial  experiments,  the  feat  has 
actually  been  performed,  and  that  by  two  distinct  methods.  In 
Fizeau's  experiments  the  distance  between  the  two  experimental 
stations  was  about  5|  miles.  In  Foucault's  experiments  the  whole 
apparatus  was  contained  in  one  room,  and  the  movement  of  light 
within  this  room  served  to  determine  the  velocity. 

We  will  first  describe  Fizeau's  experiment. 

941.  Fizeau's  Experiment. — Imagine  a  source  of  light  placed  di- 
rectly in  front  of  a  plane  mirror,  at  a  great  distance.     The  mirror 
will  send  back  a  reflected  beam  along  the  line  of  the  incident  beam, 
and  an  observer  stationed  behind  the  source  will  see  its  image  in  the 
mirror  as  a  luminous  point. 

Now  imagine  a  toothed-wheel,  with  its  plane  perpendicular  to  the 
path  of  the  beam,  revolving  uniformly  in  front  of  the  source,  in 
such  a  position  that  its  teeth  pass  directly  between  the  source  of 
light  and  the  mirror.  The  incident  beam  will  be  stopped  by  the 
teeth,  as  they  successively  come  up,  but  will  pass  through  the  spaces 
between  them.  Now  the  velocity  of  the  wheel  may  be  such  that 
the  light  which  has  thus  passed  through  a  space  shall  be  reflected 
back  from  the  mirror  just  in  time  to  meet  a  tooth  and  be  stopped. 
In  this  case  it  will  not  reach  the  observer's  eye,  and  the  image  may 
thus  become  permanently  invisible  to  him.  From  the  velocity  of 
the  wheel,  and  the  number  of  its  teeth,  it  will  be  possible  to  com- 
pute the  time  occupied  by  the  light  in  travelling  from  the  wheel  to 
the  mirror,  and  back  again.  If  the  velocity  of  the  wheel  is  such 
that  the  light  is  sometimes  intercepted  on  its  return,  and  sometimes 
allowed  to  pass,  the  image  will  appear  steadily  visible,  in  conse- 
quence of  the  persistence  of  impressions  on  the  retina,  but  with  a 


956  PROPAGATION    OF   LIGHT. 

loss  of  brightness  proportioned  to  the  time  that  the  light  is  inter- 
cepted. The  wheel  employed  by  Fizeau  had  720  teeth,  the  distance 
between  the  two  stations  was  8663  metres,  and  12'6  revolutions  per 
second  produced  disappearance  of  the  image.  The  width  of  the 
teeth  being  equal  to  the  width  of  the  spaces,  the  time  required  to 
turn  through  the  width  of  a  tooth  was  \  X  -^-^  X  y^-.-g-  of  a  second, 
that  is  Ts^-rf  of  a  second. 

In  this  time  the  light  travelled  a  distance  of  2  x  8663  =  17326 
metres.  The  distance  traversed  by  light  in  a  second  would  therefore 
be  17,326  X  18,144  =  314,262,944  metres.  This  determination  of  M. 
Fizeau's  is  believed  to  be  somewhat  in  excess  of  the  truth. 

A  double  velocity  of  the  wheel  would  allow  the  reflected  beam  to 
pass  through  the  space  succeeding  that  through  which  the  incident 
beam  had  passed;  a  triple  velocity  would  again  produce  total  eclipse, 
and  so  on.  Several  independent  determinations  of  the  velocity  of 
light  may  thus  be  obtained. 


Fig.  649. -Fizeau's  Experiment 

Thus  far,  we  have  merely  indicated  the  principle  of  calculation. 
It  will  easily  be  understood  that  special  means  were  necessary  to 
prevent  scattering  of  the  light,  and  render  the  image  visible  at  so 
great  a  distance.  Fig.  649  will  serve  to  give  an  idea  of  the  apparatus 
actually  employed. 

A  beam  of  light  from  a  lamp,  after  passing  through  a  lens,  falls 
on  a  plate  of  unsilvered  glass  M,  placed  at  an  angle  of  45°,  by 
which  it  is  reflected  along  the  tube  of  a  telescope;  the  object- 
glass  of  the  telescope  is  so  adjusted  as  to  render  the  rays  parallel 
on  emergence,  and  in  this  condition  they  traverse  the  interval 


VELOCITY    OF  LIGHT.  957 

between  the  two  stations.  At  the  second  station  they  are  collected 
by  a  lens,  which  brings  them  to  a  focus  on  the  surface  of  a  mirror, 
which  sends  them  back  along  the  same  course  by  which  they 
came.  A  portion  of  the  light  thus  sent  back  to  the  glass  plate  M 
passes  through  it,  and  is  viewed  by  the  observer  through  an  eye- 
piece. 

The  wheel  R  is  driven  by  clock-work.    Figs.  650,  651,  652  respec- 


Fig.  650. -Wheel  at  Rest  Fig.  651.— Total  Eclipse.  Fig.  652.— Partial  Eclipse. 

tively  represent  the  appearance  of  the  luminous  point  as  seen  between 
the  teeth  of  the  wheel  when  not  revolving,  the  total  eclipse  produced 
by  an  appropriate  speed  of  rotation,  and  the  partial  eclipse  produced 
by  a  different  speed. 

More  recently  M.  Cornu  has  carried  out  an  extensive  series  of 
experiments  on  the  same  plan,  with  more  powerful  appliances,  the 
distance  between  the  two  stations  being  23  kilometres,  and  the 
extinctions  being  carried  to  the  21st  order.  His  result  is  that  the 
velocity  of  light  (in  millions  of  metres  per  second)  is  300'33  in  air, 
or  300 '4  in  vacuo. 

942.  Foucault's  Experiment. — Foucault  employed  the  principle  of 
the  rotating  mirror,  first  adopted  by  Wheatstone  in  his  experiments 
on  the  duration  of  the  electric  spark  and  the  velocity  of  electricity 
(§  591,  636).  The  following  was  the  construction  of  his  original 
apparatus: — 

A  beam  of  light  enters  a  room  by  a  square  hole,  which  has  a  fine 
platinum  wire  stretched  across  it,  to  serve  as  a  mark;  it  is  then 
concentrated  by  an  achromatic  lens,  and,  before  coming  to  a  focus, 
falls  upon  a  plane  mirror,  revolving  about  an  axis  in  its  own  plane. 
In  one  part  of  the  revolution  the  reflected  beam  is  directed  upon  a 
concave  mirror,  whose  centre  of  curvature  is  in  the  axis  of  rotation, 
so  that  the  beam  is  reflected  back  to  the  revolving  mirror,  and 


958 


PROPAGATION   OF   LIGHT. 


thence  back  to  the  hole  at  which  it  first  entered.  Before  reaching 
the  hole,  it  has  to  traverse  a  sheet  of  glass,  placed  at  an  angle  of  45°, 
which  reflects  a  portion  of  it  towards  the  observer's  eye;  and  the 
image  which  it  forms  (an  image  of  the  platinum  wire)  is  viewed 
through  a  powerful  eye-piece.  The  image  is  only  formed  during  a 
small  part  of  each  revolution;  but  when  30  turns  are  made  per  second, 
the  appearance  presented,  in  consequence  of  the  persistence  of  im- 
pressions, is  that  of  a  permanent  image  occupying  a  fixed  position. 
When  the  speed  is  considerably  greater,  the  mirror  turns  through 
a  sensible  angle  while  the  light  is  travelling  from  it  to  the  concave 
mirror  and  back  again,  and  a  sensible  displacement  of  the  image  is 
accordingly  observed.  The  actual  speed  of  rotation  was  from  700  to 
800  revolutions  per  second.1 


Fig.  653.— Foucault's  Experiment. 

On  interposing  a  tube  filled  with  water  between  the  two  mirrors, 
it  was  found  that  the  displacement  was  increased,  showing  that  a 
longer  time  was  occupied  in  traversing  the  water  than  in  traversing 
the  same  length  of  air. 

This  result,  as  we  shall  have  occasion,  to  point  out  later,  is  very 
important  as  confirming  the  undulatory  theory  and  disproving  the 
emission  theory  of  light. 

In  Fig.  653,  a  is  the  position  of  the  platinum  wire,  L  is  the 
achromatic  lens,  m  the  revolving  mirror,  c  the  axis  of  revolution,  M 

1  It  was  found  that,  at  this  high  speed,  the  amalgam  at  the  back  of  ordinary  looking- 
glasses  was  driven  off  by  centrifugal  force.  The  mirror  actually  employed  was  silvered  in 
front  with  real  silver. 


VELOCITY   OF  LIGHT.  959 

the  concave  mirror,  a'  the  image  of  the  platinum  wire,  displaced  from 
a  in  virtue  of  the  rotation  of  the  mirror;  a,  a'  images  of  a,  a,  formed 
by  the  glass  plate  g,  and  viewed  through  the  eye-piece  0. 

M'  is  a  second  concave  mirror,  at  the  same  distance  as  M  from  the 
revolving  mirror;  T  is  a  tube  filled  with  water,  and  having  plane 
glass  ends,  and  L'  a  lens  necessary  for  completing  the  focal  adjust- 
ment; a"  and  o"  are  the  images  formed  by  the  light  which  has  tra- 
versed the  water.1 

Foucault's  experiment,  as  thus  described,  was  performed  in  1850 
very  shortly  after  that  of  Fizeau,  and  was  mainly  designed  for  giving 
a  relative  determination  of  the  velocities  in  air  and  water.  Foucault 
subsequently  adapted  it  to  absolute  measurement,  and  determined 
the  velocity  in  air  to  be  298  million  metres  per  second. 

943.  Later  Determinations. — Captain  Michelson  of  the  United 
States  navy  carried  out  in  1879  and  1882  two  excellent  series  of 
experiments,  in  which  the  lens  L  was  placed  not  between  the  slit 
and  the  revolving  mirror  but  between  the  revolving  and  the  fixed 
mirror,  in  such  a  position  that  the  sum  of  the  distances  of  the  slit 
and  lens  from  the  revolving  mirror  was  a  very  little  greater  than 
the  focal  length  of  the  lens.  The  image  of  the  slit  was  accordingly 
formed  at  a  very  great  distance  on  the  other  side  of  the  lens,  and  it 
was  at  this  distance  that  the  fixed  mirror  M  was  placed.  The  focal 
length  of  the  lens  was  150  ft.,  and  the  distance  between  the  two 
mirrors  nearly  2000  ft.  The  measured  deviation  of  the  image  of 

1  The  distances  are  such  that  La  and  Lc  +  cM  are  conjugate  focal  distances  with 
respect  to  the  lens  L.  An  image  of  the  wire  a  is  thus  formed  at  M,  and  an  image  of  this 
image  is  formed  at  a,  the  mirror  being  supposed  stationary ;  and  this  relation  holds  not 
only  for  the  central  point  of  the  concave  mirror,  but  for  any  part  of  it  on  which  the  light 
may  happen  to  fall  at  the  instant  considered. 

Let  I  denote  the  distance  cM  between  the  revolving  and  the  fixed  mirror,  I'  the  distance 
cL  of  the  revolving  mirror  from  the  centre  of  the  lens,  r  the  distance  aL  of  the  platinum 
wire  from  the  centre  of  the  lens,  n  the  number  of  revolutions  per  second,  V  the  space  tra- 
versed by  light  in  a  second,  t  the  time  occupied  by  light  in  travelling  from  one  mirror  to 
the  other  and  back,  6  the  angle  turned  by  the  mirror  in  this  time,  and  8  the  angle  sub- 
tended at  the  centre  of  the  lens  by  the  distance  a  a'  between  the  wire  and  its  displaced  image. 

2  I  6  4  vrn I 

Then  obviously  t=  y,  but  also  t-^^',  hence  V-     ff     . 

Now  the  distance  between  the  two  images  (corresponding  to  a,  a'  respectively)  at  the 
back  of  the  revolving  mirror  is  (I  + 1')  S,  and  is  also  2  61  (§  964).  Hence  e=^—~^,  and 

V=  -j — jf- .  The  observed  distance  a  a'  between  the  two  images  is  equal  to  the  distance 
between  a,  a',  that  is  to  r  8.  Calling  this  distance  d,  we  have  finally, 


9  GO  PROPAGATION   OF  LIGHT. 

the  slit  from  the  slit  itself  was  due  to  the  angle  through  which  the 
mirror  turned  while  light  travelled  over  twice  this  distance,  or  nearly 
4000  ft.;  and  the  distance  of  the  slit  from  the  mirror  being  about 
30  ft.,  the  deviation  of  the  image  from  the  slit  amounted  to  more 
than  133  millimetres,  whereas  the  deviation  obtained  by  Foucault 
was  less  than  1  millimetre.  The  velocity  in  vacuo  finally  deduced 
by  Michelson  was  299'9l  from  the  first  series,  and  299'85  from  the 
second  series. 

A  still  better  determination  was  made  in  1882  by  Professor  New- 
comb  of  the  United  States  Naval  Observatory,  Washington,  the 
distance  between  the  revolving  and  the  fixed  mirror  being  in  some 
of  the  observations  2550  metres,  and  in  others  3720  metres.  The 
method  employed  was  in  principle  the  same  as  in  Foucault's  original 
experiment.  The  revolving  mirror  was  four-sided,  like  that  in 
Fig.  639,  p.  937,  but  made  of  polished  steel,  and  driven  by  a  blast  of 
air  impinging  on  vanes.  The  speed  of  revolution  was  measured  by 
a  self-recording  apparatus  which,  by  breaking  an  electric  current, 
made  a  mark  once  in  every  28  revolutions  upon  paper,  on  which  a 
mark  was  also  made  every  second  by  a  chronograph. 

The  source  of  light  was  a  slit  illuminated  by  the  rays  of  the  sun 
reflected  from  a  heliostat.  The  slit  was  in  the  principal  focus  of  a 
collimating  lens,  so  that  a  parallel  beam  of  light  fell  upon  the  revolv- 
ing mirror,  and  was  reflected  to  and  from  the  distant  station.  On  its 
return  it  was  reflected  by  the  revolving  mirror  into  an  observing 
telescope  furnished  with  two  parallel  wires  near  together  in  the 
focus  of  its  eye-piece,  between  which  the  image  was  made  to  fall. 
This  telescope  was  so  mounted  that,  while  always  directed  centrally 
on  the  revolving  mirror,  it  could  be  moved  through  about  4°  on  each 
side  of  the  position  of  no  deviation;  and  in  actual  use  it  was  moved 
into  such  a  position  that  the  displaced  image  of  the  slit  could  be  kept 
steadily  between  the  two  parallel  wires,  the  regulation  of  the  dis- 
placement being  effected  by  means  of  two  cords  which  governed  the 
blast  of  air.  The  deviation  amounted  in  some  of  the  experiments  to 
3°  on  each  side,  the  arrangements  being  such  that  the  mirror  could 
be  turned  either  way.  The  result  deduced  was  a  velocity  m  vacuo 
of  299-86  million  metres  per  second  (about  186,300  miles  per  second), 
which  may  be  accepted  as  the  best  determination  yet  made. 

944.  Velocity  of  Light  deduced  from  Observations  of  the  Eclipses  of 
Jupiter's  Satellites. — The  fact  that  light  occupies  a  sensible  time  in 
travelling  over  celestial  distances,  was  first  established  about  1675, 


VELOCITY  OF   LIGHT. 


961 


by  Roemer,  a  Danish  astronomer,  who  also  made  the  first  computa- 
tion of  its  velocity.  He  was  led  to  this  discovery  by  comparing  the 
observed  times  of  the  eclipses  of  Jupiter's  first  satellite,  as  contained 
in  records  extending  over  many  successive  years. 

The  four  satellites  of  Jupiter  revolve  nearly  in  the  plane  of  the 
planet's  orbit,  and  undergo  very  frequent  eclipse  by  entering  the 
cone  of  total  shadow  cast  by  Jupiter.  The  satellites  and  their 
eclipses  are  easily  seen,  even  with  telescopes  of  very  moderate  power; 
and  being  visible  at  the  same  absolute  time  at  all  parts  of  the  earth's 
surface  at  which  they  are  visible  at  all,  they  serve  as  signals  for 
comparing  local  time  at  different  places,  and  thus  for  determining 
longitudes.  The  first  satellite  (that  is,  the  one  nearest  to  Jupiter), 
from  its  more  rapid  motion  and  shorter  time  of  revolution,  affords 
both  the  best  and  the  most  frequent  signals.  The  interval  of  time 
between  two  successive  eclipses  of  this  satellite  is  about  42 1  hours, 
but  was  found  by  Roemer  to  vary  by  a  regular  law  according  to  the 
position  of  the  earth  with  respect  to  Jupiter.  It  is  longest  when 
the  earth  is  increasing  its  distance  from  Jupiter  most  rapidly,  and  is 
shortest  when  the  earth  is  diminish- 
ing its  distance  most  rapidly.  Start- 
ing from  the  time  when  the  earth  is 
nearest  to  Jupiter,  as  at  T,  J  (Fig. 
654),  the  intervals  between  successive 
eclipses  are  always  longer  than  the 
mean  value,  until  the  greatest  dis- 
tance has  been  attained,  as  at  T'  J', 
and  the  sum  of  the  excesses  amounts 
to  16  min.  26'6  sec.  From  this  time 
until  the  nearest  distance  is  again 
attained,  as  at  T",  J",  the  intervals  are 
always  shorter  than  the  mean,  and  the 

sum  of  the  defects  amounts  to  16  min.  26'6  sec.  It  is  evident,  then, 
that  the  eclipses  are  visible  16  m.  2 6 '6  s.  earlier  at  the  nearest  than 
at  the  remotest  point  of  the  earth's  orbit;  in  other  words,  that  this  is 
the  time  required  for  the  propagation  of  light  across  the  diameter  of 
the  orbit.  Taking  this  diameter  as  184  millions  of  miles,1  we  have 
a  resulting  velocity  of  about  186,500  miles  per  second. 

945.  Velocity  of  Light   deduced   from   Aberration. — About  fifty 

1  The  sun's  mean  distance  from  the  earth  was,  until  recently,  estimated  at  95  millions 
of  miles.     It  is  now  estimated  at  92  or  92J  millions. 
61 


Fig.  654. — Earth  and  Jupiter. 


962  PROPAGATION   OF   LIGHT. 

years  after  Roemer's  discovery,  Bradley,  the  English  astronomer, 
employed  the  velocity  of  light  to  explain  the  astronomical  pheno- 
menon called  aberration.  This  consists  in  a  regular  periodic  displace- 
ment of  the  stars  as  seen  from  the  earth,  the  period  of  the  displace- 
ment being  a  year.  If  the  direction  in  which  the  earth  is  moving 
in  its  orbit  at  any  instant  be  regarded  as  the  forward  direction 
every  star  constantly  appears  on  the  forward  side  of  its  true  place, 
so  that,  as  the  earth  moves  once  round  its  orbit  in  a  year,  each  star 
describes  in  this  time  a  small  apparent  orbit  about  its  true  place. 

The  phenomenon  is  explained  in  the  same  way  as  the  familiar 
fact,  that  a  shower  of  rain  falling  vertically,  seems,  to  a  person  run- 
ning  forwards,  to  be  coming  in  his  face.  The  relative 
motion  of  the  rain-drops  with  respect  to  his  body,  is 
found  by  compounding  the  actual  velocity  of  the 
drops  (whether  vertical  or  oblique)  with  a  velocity 
equal  and  opposite  to  that  with  which  he  runs.  Thus 
if  A  B  (Fig.  655)  represents  the  velocity  with  which 
he  runs,  and  C  A,  the  true  velocity  of  the  drops,  the 
apparent  velocity  of  the  drops  will  be  represented 
by  DA.  If  a  tube  pointed  along  A  D  moves  forward 
parallel  to  itself  with  the  velocity  A  B,  a  drop  enter- 
ing at  its  upper  end  will  pass  through  its  whole  length 
without  wetting  its  sides;  for  while  the  drop  is  falling 
along  D  B  (we  suppose  with  uniform  velocity)  the 
tube  moves  along  A  B,  so  that  the  lower  end  of  the  tube  reaches  B 
at  the  same  time  as  the  rain-drop. 

In  like  manner,  if  A  B  is  the  velocity  of  tne  earth,  and  C  A  the 
velocity  of  light,  a  telescope  must  be  pointed  along  A  D  to  see  a  star 
which  really  lies  in  the  direction  of  A  C  or  B  D  produced.  When 
the  angle  B  A  C  is  a  right  angle  (in  other  words,  when  the  star  lies 
in  a  direction  perpendicular  to  that  in  which  the  earth  is  moving), 
the  angle  CAD,  which  is  called  the  aberration  of  the  star,  is  20"*5, 
and  the  tangent  of  this  angle  is  the  ratio  of  the  velocity  of  the  earth 
to  the  velocity  of  light.  Hence  it  is  found  by  computation  that  the 
velocity  of  light  is  about  ten  thousand  times  greater  than  that  with 
which  the  earth  moves  in  its  orbit.  The  latter  is  easily  computed, 
if  the  sun's  distance  is  known,  and  is  about  18  J  miles  per  second. 
Hence  the  velocity  of  light  is  about  185,000  miles  per  second.  It 
will  be  noted  that  both  these  astronomical  methods  of  computing  the 
velocity  of  light,  depend  upon  the  knowledge  of  the  sun's  distance 


PHOTOMETRY.  963 

from  the  earth,  and  that,  if  this  distance  is  overestimated,  the  com- 
puted velocity  of  light  will  be  too  great  in  the  same  ratio. 

Conversely,  the  velocity  of  light,  as  determined  by  Foucault's 
method,  can  be  employed  in  connection  either  with  aberration  or 
the  eclipses  of  the  satellites,  for  computing  the  sun's  distance;  and 
the  first  correct  determination  of  the  sun's  distance  was,  in  fact,  that 
deduced  by  Foucault  from  his  own  results. 

946.  Photometry. — Photometry  is  the  measurement  of  the  relative 
amounts  of  light  emitted  by  different  sources.     The  methods  em- 
ployed for  this  purpose  all  consist  in  determinations  of  the  relative 
distances  at  which  two  sources  produce  equal  intensities  of  illumina- 
tion.    The  eye  would  be  quite  incompetent  to  measure  the  ratio  of 
two  unequal  illuminations;  but  a  pretty  accurate  judgment  can  be 
formed  as  to  equality  or  inequality  of  illumination,  at  least  when  the 
surfaces  compared  are  similar,  and  the  lights  by  which  they  are  illu- 
minated are  of  the  same  colour.    The  law  of  inverse  squares  is  always 
made  the  basis  of  the  resulting  calculations;  and  this  law  may  itself 
be  verified  by  showing  that  the  illumination  produced  by  one  candle 
at  a  given  distance  is  equal  to  that  produced  by  four  candles  at  a 
distance  twice  as  great. 

947.  Bouguer's  Photometer. — Bouguer's  photometer  consists  of  a 


Fig.  656.—  Bouguer's  Photometer. 


semi-transparent  screen,  of  white  tissue  paper,  ground  glass,  or  thin 
white  porcelain,  divided  into  two  parts  by  an  opaque  partition  at 
right  angles  to  it.  The  two  lamps  which  are  to  be  compared  are 


964  PROPAGATION   OF   LIGHT. 

placed  one  on  each  side  of  this  partition,  so  that  each  of  them  illu- 
minates one-half  of  the  transparent  screen.  The  distances  of  the 
two  lamps  are  adjusted  until  the  two  portions  of  the  screen,  as  seen 
from  the  back,  appear  equally  bright.  The  distances  are  then  mea- 
sured, and  their  squares  are  assumed  to  be  directly  proportional  to 
the  illuminating  powers  of  the  lamps. 

948.  Rumford's  Photometer. — Rumford's  photometer  is  based  on 
the  comparison  of  shadows.  A  cylindric  rod  is  so  placed  that  each  of 
the  two  lamps  casts  a  shadow  of  it  on  a  screen;  and  the  distances 


657.-Rumlord's  Photometer. 


are  adjusted  until  the  two  shadows  are  equally  dark.  As  the  shadow 
thrown  by  one  lamp  is  illuminated  by  the  other  lamp,  the  compari- 
son of  shadows  is  really  a  comparison  of  illuminations. 

949.  Foucault's  Photometer. — The  two  photometers  just  described 
are  alike  in  principle.  In  each  of  them  the  two  surfaces  compared 
are  illuminated  each  by  one  only  of  the  sources  of  light.  In  Rum- 
ford's  the  remainder  of  the  screen  is  illuminated  by  both.  In 
Bouguer's  it  consists  merely  of  an  intervening  strip  which  is  illumi- 
nated by  neither.  If  the  partition  is  movable,  the  effect  of  moving 
it  further  from  the  screen  will  be  to  make  this  dark  strip  narrower 
until  it  disappears  altogether;  and  if  it  be  advanced  still  further,  the 
two  illuminated  portions  will  overlap.  In  Foucault's  photometer 
there  is  an  adjusting  screw,  for  the  purpose  of  advancing  the  parti- 


PHOTOMETERS.  965 

tion  so  far  that  the  dark  strip  shall  just  vanish.  The  two  illuminated 
portions,  being  then  exactly  contiguous,  can  be  more  easily  and 
certainly  compared. 

950.  Bunsen's  Photometer. — In  the  instruments  above  described 
the  two  sources  to  be  compared  are  both  on  the  same  side  of  the 
screen,  and  illuminate  different  portions  of  it.     Bunsen  introduced 
the  plan  of  placing  the  sources  on  opposite  sides  of  the  screen,  and 
making  the  screen  consist  of  two  parts,  one  of  them  more  translucent 
than  the  other.     In  his  original  pattern  the  screen  was  a  sheet  of 
white  paper,  with  a  large  grease  spot  in  the  centre.    In  Dr.  Letheby's 
pattern  it  is  composed  of  three  sheets  of  paper,  laid  face  to  face,  the 
middle  one  being  very  thin,  and  the  other  two  being  cut  away  in 
the  centre,  so  that  the  central  part  of  the  screen  consists  of  one 
thickness,  and  the  outer  part  of  three. 

When  such  a  screen  is  more  strongly  illuminated  on  one  side  than 
on  the  other,  the  more  translucent  part  appears  brighter  than  the 
less  translucent  when  seen  from  the  darker  side,  while  the  reverse 
appearance  is  presented  on  the  brighter  side.     It  is  therefore  the 
business  of  the  observer  so  to  adjust  the  distances  that  the  central 
and   circumferential   parts   ap- 
pear equally  bright.  When  they 
appear  equally  bright  from  one 
side    they    will     also     appear     A 
equally  bright  from  the  other; 

but   as   there   is  always  some  Fig.  ess. -Lethebys  Photometer, 

little  difference  of  tint,  the  ob- 
server's judgment  is  aided  by  seeing  both  sides  at  once.     This  is 
accomplished  in  Dr.  Letheby's  photometer  by  means  of  two  mirrors) 
one  for  each  eye,  as  represented  in  the  accompanying  ground-plan 
(Fig.  658). 

s  is  the  screen,  and  nm  are  the  two  mirrors,  in  which  images  s'  s' 
are  seen  by  an  observer  in  front.  The  frame  which  carries  the 
screen  and  mirrors  travels  along  a  graduated  bar  A  B,  on  which  the 
distances  of  the  screen  from  the  two  lights  are  indicated. 

In  all  delicate  photometric  observations,  the  eye  should  be  shielded 
from  direct  view  of  the  lights,  and,  as  much  as  possible,  from  all 
extraneous  lights.  The  objects  to  be  compared  should  be  brighter 
than  anything  else  in  the  field  of  view. 

951.  Photometers  for  very  Powerful  Lights. — In  comparing  two 
very  unequal  lights,  for  example,  a  powerful  electric  light  and  a 


966  PROPAGATION   OF   LIGHT. 

standard  candle,  it  is  scarcely  possible  to  obtain  an  observing-roora 
long  enough  for  a  direct  comparison  by  any  of  the  above  methods. 
To  overcome  this  difficulty  a  lens  (either  convex  or  concave),  of 
short  focal  length,  may  be  employed  to  form  an  image  of  the  more 
powerful  source  near  its  principal  focus.  Then  all  the  light  which 
this  source  sends  to  the  lens  may  be  regarded  as  diverging  from  the 
image  and  filling  a  solid  angle  equal  to  that  which  the  lens  subtends 
at  the  image.  In  other  words,  the  illuminations  of  the  lens  itself 
due  to  the  source  and  the  image  are  equal.  Hence,  if  S  and  I  are 
the  distances  of  the  source  and  image  from  the  lens,  the  image  is 
weaker  than  the  source  in  the  ratio  of  I2  to  S2,  and  a  direct  com- 
parison can  be  made  between  the  light  from  the  image  and  that 
from  a  standard  candle.  Thus,  if  a  screen  at  distance  D  from  the 
image  has  the  same  illumination  from  the  image  as  from  a  candle 

T)2 

at  distance  C  on  the  other  side,  the  image  is  equal  to  ^  candles,  and 

T\2    Q2 

the  source  itself  to  ^  p  candles.     A  correction  must,  however,  be 

applied  to  this  result  for  the  light  lost  by  reflection  at  the  surfaces 
of  the  lens. 


CHAPTER    LXVIII. 


REFLECTION   OF   LIGHT. 


952.  Reflection.— If  a  beam  of  the  sun's  rays  AB  (Fig.  659)  be 
admitted  through  a  small  hole  in  the  shutter  of  a  dark  room,  and 
allowed  to  fall  on  a  polished  plane  surface,  it  will  be  seen  to  continue 
its  course  in  a  different  direction  B  C.  This  is  an  example  of  reflec- 


Fig.  659.— Reflection  of  Light. 

tion.  A  B  is  called  the  incident  beam,  and  B  C  the  reflected  beam. 
The  angle  A  B  D  contained  between  an  incident  ray  and  the  normal 
is  called  the  angle  of  incidence;  and  the  angle  CBD  contained 
between  the  corresponding  reflected  ray  and  the  normal  is  called 
the  angle  of  reflection.  The  plane  ABD  containing  the  incident 
ray  and  the  normal  is  called  the  plane  of  incidence. 

953.  Laws  of  Reflection. — The  reflection  of  light  from  polished 
surfaces  takes  place  according  to  the  following  laws: — 

1.  The  reflected  ray  lies  in  the  plane  of  incidence. 


968 


REFLECTION   OF  LIGHT. 


2.  The  angle  of  reflection  is  equal  to  the  angle  of  incidence. 
These  laws  may  be  verified  by  means  of  the  apparatus  represented 
in  Fig.  660.     A  vertical  divided  circle  has  a  small  polished  plate 
fixed  at  its  centre,  at  right  angles  to  its  plane, 
and  two  tubes  travelling  on  its  circumference 
with  their  axes  always  directed  towards  the 
centre.    The  zero  of  the  divisions  is  the  highest 
point  of  the  circle,  the  plate  being  horizontal. 

A  source  of  light,  such  as  the  flame  of  a 
candle,  is  placed  so  that  its  rays  shine  through 
one  of  the  tubes  upon  the  plate  at  the  centre. 
As  the  tubes  are  blackened  internally,  no  light 
passes  through  except  in  a  direction  almost 
Precisel7  parallel  to  the  axis  of  the  tube.  The 
observer  then  looks  through  the  other  tube, 
and  moves  it  along  the  circumference  till  he  finds  the  position  in 
which  the  reflected  light  is  visible  through  it.  On  examining  the 
graduations,  it  will  be  found  that  the  two  tubes  are  at  the  same 
distance  from  the  zero  point,  on  opposite  sides.  Hence  the  angles 
of  incidence  and  reflection  are  equal.  Moreover  the  plane  of  the 
circle  is  the  plane  of  incidence,  and  this  also  contains  the  reflected 
rays.  Both  the  laws  are  thus  verified. 

954.  Artificial  Horizon. — These  laws  furnish  the  basis  of  a  method 
of  observation  which  is  frequently  employed  for  determining  the 
altitude  of  a  star,  and  which,  by  the  consistency  of  its  results,  fur- 
nishes a  very  rigorous  proof  of  the  laws. 

A  vertical  divided  circle  (Fig.  661)  is  set  in  a  vertical  plane  by 
proper  adjustments.  A  telescope  movable  about  the  axis  of  the 
circle  is  pointed  to  a  particular  star,  so  that  its  line  of  collimation 
I'  S'  passes  through  the  apparent  place  of  the  star.  Another  tele- 
scope,1 similarly  mounted  on  the  other  side  of  the  circle,  is  directed 
downwards  along  the  line  I'R  towards  the  image  of  the  star  as  seen 
in  a  trough  of  mercury  I.  Assuming  the  truth  of  the  laws  of  reflec- 
tion as  above  stated,  the  altitude  of  the  star  is  half  the  angle  between 
the  directions  of  the  two  telescopes;  for  the  ray  S I  from  -fche  star  to 
the  mercury  is  parallel  to  the  line  S'  I',  by  reason  of  the  excessively 
great  distance  of  the  star;  and  since  the  rays  SI,  IR  are  equally 
inclined  to  the  normal  IN,  which  is  a  vertical  line,  the  lines  I'S,  I'R 
are  also  equally  inclined  to  the  vertical,  or,  what  is  the  same  thing, 

1  In  practice,  a  single  telescope  usually  serves  for  both  observations. 


ARTIFICIAL  HORIZON.  969 

are  equally  inclined  to  a  horizontal  plane.      A  reflecting  surface 
of  mercury  thus  used  is  called  a  mercury  horizon,  or  an  artificial 


Fig.  661.- Artificial  Horizon. 

horizon.  Observations  thus  made  give  even  more  accurate  results 
than  those  in  which  the  natural  horizon  presented  by  the  sea  is  made 
the  standard  of  reference. 

955.  Irregular  Reflection. — The  reflection  which  we  have  thus  far 
been  discussing  is  called  regular  reflection.  It  is  more  marked  as 
the  reflecting  surface  is  more  highly  polished,  and  (except  in  the 
case  of  metals)  as  the  incidence  is  more  oblique.  But  there  is  an- 
other kind  of  reflection,  in  virtue  of  which  bodies,  when  illuminated, 
send  out  light  in  all  directions,  and  thus  become  visible.  This  is 
called  irregular  reflection  or  diffusion.  Regular  reflection  does  not 
render  the  reflecting  body  visible,  but  exhibits  images  of  surrounding 
objects.  A  perfectly  reflecting  mirror  would  be  itself  unseen,  and 


970 


REFLECTION   OF  LIGHt. 


actual  mirrors  are  only  visible  in  virtue  of  the  small  quantity  of 
diffused  light  which  they  usually  emit.  The  transformation  of  in- 
cident into  diffused  light  is  usually  selective;  so  that,  though  the 
incident  beam  may  be  white,  the  diffused  light  is  usually  coloured. 
The  power  which  a  body  possesses  of  making  such  selection  consti- 
tutes its  colour. 

The  word  reflection  is  often  used  by  itself  to  denote  what  we  have 
here  called  regular  reflection,  and  we  shall  generally  so  employ  it. 

956.  Mirrors. — The  mirrors  of  the  ancients  were  of  metal,  usually 
of  the  compound  now  known  as  speculum-metal.  Looking-^/ lasses 
date  from  the  twelfth  century.  They  are  plates  of  glass,  coated  at 
the  back  with  an  amalgam  of  quicksilver  and  tin,  which  forms  the 
reflecting  surface.  This  arrangement  has  the  great  advantage  of 
excluding  the  air,  and  thus  preventing  oxidation.  It  is  attended, 
however,  with  the  disadvantage  that  the  surface  of  the  glass  and 
the  surface  of  the  amalgam  form  two  mirrors;  and  the  superposition 
of  the  two  sets  of  images  produces  a  confusion  which  would  be  in- 
tolerable in  delicate  optical  arrangements.  The  mirrors,  or  specula 
as  they  are  called,  of  reflecting  telescopes  are  usually  made  of  specu- 
lum-metal, which  is  a  bronze  composed  of  about  32  parts  of  copper 
to  15  of  tin.  Lead,  antimony,  and  arsenic  are  sometimes  added.  Of 
late  years  specula  of  glass  coated  in  front  with  real  silver  have  been 
extensively  used;  they  are  known  as  silvered  specula.  A  coating 
of  platinum  has  also  been  tried,  but  not  with  much  success.  The 
mirrors  employed  in  optics  are  usually 
either  plane  or  spherical 

957.  Plane  Mirrors. — By  a  plane 
mirror  we  mean  any  plane  reflecting 
/^"surface.  Its  effect,  as  is  well  known, 
is  to  produce, behind  the  mirror, images 
exactly  similar,  both  in  form  and  size, 
to  the  real  objects  in  front  of  it.  This 
phenomenon  is  easily  explained  by  the 
laws  of  reflection. 

Let  M  N  (Fig.  662)  be  a  plane  mir- 
ror, and  S  a  luminous  point.  Rays 
SI,  SI',  SI"  proceeding  from  this  point 

give  rise  to  reflected  rays  I  O,  TO',  FO";  and  each  of  these,  if 
produced  backwards,  will  meet  the  normal  S  K  in  a  point  S',  which 
is  at  the  same  distance  behind  the  mirror  that  S  is  in  front  of 


Fig.  662. -Plane  Mirror. 


IMAGE   IN  PLANE  MIRROR. 


971 


Fig.  663.— Image  of  Caudle. 


it.1  The  reflected  rays  have  therefore  the  same  directions  as  if  they 
had  come  from  S',  and  the  eye  receives  the  same  impression  as  if 
S'  were  a  luminous  point. 

Fig.  663  represents  a  pencil  of  rays  emitted  by  the  highest  point 
of  a  candle-flame,  and  re- 
flected from  a  plane  mir- 
ror to  the  eye  of  an  ob- 
server. The  reflected  rays 
are  divergent  (like  the  in- 
cident rays),  and  if  pro- 
duced backwards  would 
meet  in  a  point,  which  is 
the  position  of  the  image 
of  the  top  of  the  flame. 

As  an  object  is  made  up 
of  points,  these  principles 
show  that  the  image  of 
an  object  formed  by  a  plane  mirror  must  be  equal  to  the  object, 
and  symmetrically  situated  with  respect  to  the  plane  of  the  mirror. 
For  example,  if  AB  (Fig.  664)  is 
an  object  in  front  of  the  mirror, 
an  eye  placed  at  O  will  see  the 
image  of  the  point  A  at  A',  the 
image  of  B  at  B',  and  so  on  for 
all  the  other  points  of  the  ob- 
ject. The  position  of  the  image 
A'B'  depends  only  on  the  posi- 
tions of  the  object  and  of  the 
mirror,  and  remains  stationary 
as  the  eye  is  moved  about.  It 
is  possible,  however,  to  find 
positions  from  which  the  eye 
will  not  see  the  image  at  all,  the 
conditions  of  visibility  being 

the  same  as  if  the  image  were  a  real  object,  and  the  mirror  were  an 
opening  through  which  it  could  be  seen. 

The  images  formed  by  a  plane  mirror  are  erect.     They  are  not 
however  exact  duplicates  of  the  objects  from  which  they  are  formed, 

1  This  is  evident  from  the  comparison  of  the  two  triangles  S  K  I,  S'  K  I,  bearing  in  mind 
that  the  angle  N I S  is  equal  to  the  alternate  angle  I  S  K,  and  N 1 0  to  K  S'  L 


Fig.  C04.—  Incident  and  Reflected  Pencils. 


972 


REFLECTION    OF   LIGHT. 


but  differ  from  them  precisely  in  the  same  way  as  the  left  foot  or 
hand  differs  from  the  right.  The  image  of  a  printed  page  is  like  the 

appearance  of  the  page  as 
seen  through  the  paper  from 
the  back,  or  like  the  type 
from  which  the  page  was 
printed. 

958.  Images  of  Images. — 
When  rays  from  a  luminous 
point  m  have  been  reflected 
from  a  mirror  A  B  (Fig.  665), 
their  subsequent  course  is 

the  same  as  if  they  had  come  from  the  image  m'  at  the  back  of 
the  mirror.  Hence,  if  they  fall  upon  a  second  mirror  C  D,  an  image 
m"  of  the  first  image  will  be  formed  at  the  back  of  the  second 
mirror.  If,  after  this,  they  undergo  a  third  reflection,  an  image 
of  m"  will  be  formed,  and  so  on  indefinitely.  The  figure  shows  the 
actual  paths  of  two  rays  mirs,  m i'r' s'.  They  diverge  first  from 


Fig.  605. — Reflection  from  two  Mirror*. 


Fig.  666.— Parallel  Mirrors. 


m,  then  from  m,  and  lastly  from  m".  This  is  the  principle  of  the 
multiple  images  formed  by  two  or  more  mirrors,  as  in  the  following 
experiments. 

959.  Parallel  Mirrors. — Let  an  object  0  be  placed  between  two 


IMAGES   OF  IMAGES. 


973 


parallel  mirrors  which  face  each  other,  as  in  Fig  G6C.  The  first 
reflections  will  form  images  al  ov  The  second  reflections  will  form 
images  «2o2  of  the  first  images;  and  the  third  reflections  will  form 
images  a3  o3  of  the  second  images.  The  figure  represents  an  eye  re- 
ceiving the  rays  which  form  the  third  images,  and  shows  the  paths 
which  these  rays  have  taken  in  their  whole  course  from  the  object  0 
to  the  eye.  The  rays  by  which 
the  same  eye  sees  the  other  images 
are  omitted,  to  avoid  confusing 
the  figure.  A  long  row  of  images 
can  thus  be  seen  at  once,  becom- 
ing more  and  more  dim  as  they 
recede  in  the  distance,  inasmuch 
as  each  reflection  involves  a  loss 
of  light. 

If  the  mirrors  are  truly  parallel, 
all  the  images  will  be  ranged  in          Fig.  667. -Mirrors  at  Eight  Angles, 
one  straight  line,  which  will  be 

normal  to  the  mirrors.    If  the  mirrors  are  inclined  at  any  angle,  the 
images  will  be  ranged  on  the  circumference  of  a  circle,  whose  centre 


Fig.  668.— Mirrors  at  Eight  Angles. 


is  on  the  line  in  which  the  reflecting  surfaces  would  intersect  if  pro- 
duced. This  principle  is  sometimes  employed  as  a  means  of  adjusting 
mirrors  to  exact  parallelism. 

960.  Mirrors  at  Right  Angles.— Let  two  mirrors  0  A,  0  B  (Fig.  G67), 


974  EEFLECTION   OF  LIGHT. 

be  set  at  right  angles  to  each  other,  facing  inwards,  and  let  m  be  a 
luminous  point  placed  between  them.  Images  m  m"  will  be  formed 
by  first  reflections,  and  two  coincident  images  will  be  formed  at  m'"  by 
second  reflections.  No  third  reflection  will  occur,  for  the  point  m"', 
being  behind  the  planes  of  both  the  mirrors,  cannot  be  reflected  in 
either  of  them.  Counting  the  two  coincident  images  as  one,  and  also 
counting  the  object  as  one,  there  will  be  in  all  four  images,  placed 
at  the  four  corners  of  a  rectangle.  Fig.  608  will  give  an  idea  of  the 
appearance  actually  presented  when  one  of  the  mirrors  is  vertical 
and  the  other  horizontal.  When  both  the  mirrors  are  vertical,  an 
observer  sees  his  own  image  constantly  bisected  by  their  common 
section  in  a  way  which  appears  at  first  sight  very  paradoxical. 

961.  Mirrors  Inclined  at  60  Degrees. — A  symmetrical  distribution 

of  images  may  be  obtained  by  placing 
a  pair  of  mirrors  at  any  angle  which 
is  an  aliquot  part  of  360°.  If,  for 
example,  they  be  inclined  at  60°  to 
each  other,  the  number  of  images, 
counting  the  object  itself  as  one,  will 
be  six.  Their  position  is  illustrated 
by  Fig.  669.  The  object  is  placed  in 
the  sector  A  C  B.  The  images  formed 
by  first  reflections  are  situated  in  the 
two  neighbouring  sectors  B  C  A',  ACB'; 

Fig.  6C9.-images  in  Kaleidoscope. the  images  formed  by  second  reflec- 
tions are  in  the  sectors  B'  C  A",  A'C  B", 
and  these  yield,  by  third  reflections,  two  coincident  images  in  the 
sector  B"  C  A",  which  is  vertically  opposite  to  the  sector  A  C  B  in 
which  the  object  lies,  and  is  therefore  behind  the  planes  of  both 
mirrors,  so  that  no  further  reflection  can  occur. 

962.  Kaleidoscope. — The  symmetrical  distribution  of  images,  ob- 
tained by  two  mirrors  inclined  at  an  angle  which  is  an  aliquot  part 
of  four  right  angles,  is  the  principle  of  the  kaleidoscope,  an  optical 
toy  invented  by  Sir  David  Brewster.    It  consists  of  a  tube  containing 
two  glass  plates,  extending  along  its  whole  length,  and  inclined  at 
an  angle  of  60°.    One  end  of  the  tube  is  closed  by  a  metal  plate,  with 
the  exception  of  a  hole  in  the  centre,  through  which  the  observer 
looks  in;  at  the  other  end  there  are  two  plates,  one  of  ground  and  the 
other  of  clear  glass  (the  latter  being  next  the  eye),  with  a  number 
of  little  pieces  of  coloured  glass  lying  loosely  between  them.     These 


IMAGES   OF   IMAGES.  975 

coloured  objects,  together  with  their  images  in  the  mirrors,  form  sym- 
metrical patterns  of  great  beauty,  which  can  be  varied  by  turning  or 
shaking  the  tube,  so  as  to  cause  the  pieces  of  glass  to  change  their 
positions. 

A  third  reflecting  plate  is  sometimes  employed,  the  cross-section 
of  the  three  forming  an  equilateral  triangle.  As  each  pair  of  plates 
produces  a  kaleidoscopic  pattern,  the  arrangement  is  nearly  equiva- 
lent to  a  combination  of  three  kaleidoscopes. 

The  kaleidoscope  is  capable  of  rendering  important  aid  to  designers. 


Fig.  670.— Kaleidoscopic  Pattern. 

Fig.  670  represents  a  pattern  produced  by  the  equilateral  arrange- 
ment of  three  reflectors  just  described. 

963.  Pepper's  Ghost. — Many  ingenious  illusions  have  been  con- 
trived, depending  on  the  laws  of  reflection  from  plane  surfaces.  We 
shall  mention  two  of  the  most  modern. 

In  the  magic  cabinet,  there  are  two  vertical  mirrors  hinged  at  the 
twa  back  corners  of  the  cabinet,  and  meeting  each  other  at  a  right 
angle,  so  as  to  make  angles  of  45°  with  the  sides,  and  also  with  the 
back.  A  spectator  seeing  the  images  of  the  two  sides,  mistakes 
them  for  the  back,  which  they  precisely  resemble;  and  performers 
may  be  concealed  behind  the  mirrors  when  the  cabinet  appears 
empty.  If  one  of  the  persons  thus  concealed  raises  his  head  above 
the  mirrors,  it  will  appear  to  be  suspended  in  mid-air  without  a  body. 


976 


REFLECTION   OF   LIGHT. 


The  striking  spectral  illusion  known  as  Peppers  Ghost  is  produced 
by  reflection  from  a  large  sheet  of  unsilvered  glass,  which  is  so  ar- 
ranged that  the  actors  on  the  stage  are  seen  through  it,  while  other 
actors,  placed  in  strong  illumination,  and  out  of  the  direct  view  of  the 
spectators,  are  seen  by  reflection  in  it,  and  appear  as  ghosts  on  the  stage. 

964.  Deviation  produced  by  Rotation  of  Mirror.— Let  AB  (Fig. 
671)  represent  a  mirror  perpendicular  to  the  plane  of  the  paper,  and 

capable  of  being  rotated  about  an  axis 
through  C,  also  perpendicular  to  the  paper; 
and  let  1C  represent  an  incident  ray. 
When  the  mirror  is  in  the  position  AB, 
perpendicular  to  I  C,  the  ray  will  be  re- 
flected directly  back  upon  its  course;  but 
when  the  mirror  is  turned  through  the 
acute  angle  A  C  A',  the  reflected  ray  will 
take  the  direction  OR,  making  with  the 
Fig.6n.-EffectofrotatingaMirror.  normal  £  N  an  angle  N  C  R,  equal  to  the 

angle  of  incidence  NCI.  The  deviation  I C  R  of  the  reflected  ray, 
produced  by  rotating  the  mirror,  is  therefore  double  of  the  angle  ICN 
or  A  C  A',  through  which  the  mirror  has  been  turned;  and  if,  starting 

from  the  position  A'B',  we  turn  the 
mirror  through  a  further  angle  0,  the 
reflected  ray  C  R  will  be  turned 
through  a  further  angle  2  9.  It  thus 
appears,  that,  when  a  plane  mirror 
is  rotated  in  the  plane  of  incidence, 
the  direction  of  the  reflected  ray  is 
changed  by  double  the  angle  through 
which  the  mirror  is  turned.  Con- 
versely, if  we  assign  a  constant  direc- 
tion C I  to  the  reflected  ray,  the 
direction  of  the  incident  ray  RC  must 
vary  by  double  the  angle  through 
which  the  mirror  is  turned. 

965.  Hadley's  Sextant. — The  above  principle  is  illustrated  in  the 
nautical  instrument  called  the  sextant  or  quadrant,  which  was  in- 
vented by  Newton,  and  reinvented  by  Hadley.     It  serves  for  mea- 
suring the  angle  between  any  two  distant  objects  as  seen  from  the 
station  occupied  by  the  observer.    Its  essential  parts  are  represented 
in  Fig.  672. 


Fig.  672.  -Sextant. 


SEXTANT.  977 

It  has  two  plane  mirrors  A,  B,  one  of  which,  A,  is  fixed  to  the 
frame  of  the  instrument,  and  is  only  partially  silvered,  so  that  a  dis- 
tant object  in  the  direction  AH  can  be  seen  through  the  unsilvered 
part.  The  other  mirror  B  is  mounted  on  a  movable  arm  B  I,  which 
carries  an  index  I,  traversing  a  graduated  arc  P  Q.  When  the  two 
mirrors  are  parallel,  the  index  is  at  P,  the  zero  of  the  graduations, 
and  a  ray  H'  B  incident  on  B  parallel  to  H  A,  will  be  reflected  first 
along  B  A,  and  then  along  A  T,  the  continuation  of  H  A.  The  ob- 
server looking  through  the  telescope  T  thus  sees,  by  two  reflections, 
the  same  objects  which  he  also  sees  directly  through  the  unsilvered 
part  of  the  mirror.  Now  let  the  index  be  advanced  through  an  angle 
0;  then,  by  the  principles  of  last  section,  the  incident  ray  SB  makes 
with  H'  B,  or  H  A,  an  angle  20.  The  angle  between  S  B  and  H  A 
would  therefore  be  given  by  reading  off  the  angle  through  which  the 
index  has  been  advanced,  and  doubling;  but  in  practice  the  arc  PQ 
is  always  graduated  on  the  principle  of  marking  half  degrees  as  whole 
ones,  so  that  the  reading  at  I  is  the  required  angle  29.  In  using  the 
instrument,  the  two  objects  which  are  to  be  observed  are  brought 
into  apparent  coincidence,  one  of  them  being  seen  directly,  and  the 
other  by  successive  reflection  from  the  two  mirrors.  This  coincidence 
is  not  disturbed  by  the  motion  of  the  ship;  but  unpractised  observers 
often  find  a  difficulty  in  keeping  both  objects  in  the  field  of  view. 
Dark  glasses,  not  shown  in  the  figure,  are  provided  for  protecting  the 
eye  in  observations  of  the  sun,  and  a  vernier  and  reading  microscope 
are  provided  instead  of  the  pointer  I. 

966.  Spherical  Mirrors. — By  a  spherical  mirror  is  meant  a  mirror 


Fig.  678.— Principal  Focus. 


whose  reflecting  surface  is  a  portion  (usually  a  very  small  portion) 
of  the  surface  of  a  sphere.  It  is  concave  or  convex  according  as  the 
inside  or  outside  of  the  spherical  surface  yields  the  reflection.  The 
centre  of  the  sphere  (C,  Fig.  673)  is  called  the  centre  of  curvature  of 


973  REFLECTION   OF  LIGHT. 

the  mirror.  If  the  mirror  has  a  circular  boundary,  as  is  usually  the 
case,  the  central  point  A  of  the  reflecting  surface  may  conveniently 
be  called  the  pole  of  the  mirror.  Centre  of  the  mirror  is  an  ambigu- 
ous phrase,  being  employed  sometimes  to  denote  the  pole,  and  some- 
times the  centre  of  curvature.  The  line  A  C  is  called  the  principal 
axis  of  the  mirror,  and  any  other  straight  line  through  C  which 
meets  the  mirror  is  called  a  secondary  axis. 

When  the  incident  rays  are  parallel  to  the  principal  axis,  the  re- 
flected rays  converge  to  a  point  F,  which  is  called  the  principal 
focus.  This  law  is  rigorously  true  for  parabolic  mirrors  (generated 
by  the  revolution  of  a  parabola  about  its  principal  axis).  For  sphe- 
rical mirrors  it  is  only  approximately  true,  but  the  approximation 
is  very  close  if  the  mirror  is  only  a  very  small  portion  of  an  entire 
sphere.  In  grinding  and  polishing  the  specula  of  large  reflecting 
telescopes,  the  attempt  is  made  to  give  them,  as  nearly  as  possible, 
the  parabolic  form.  Parabolic  mirrors  are  also  frequently  employed 


Fig.  674.— Theory  of  Conjugate  Foci 

to  reflect,  in  a  definite  direction,  the  rays  of  a  lamp  placed  at  the 
focus. 

Rays  reflected  from  the  circumferential  portion  of  a  spherical  mir- 
ror are  always  too  convergent  to  concur  exactly  with  those  reflected 
from  the  central  portion.  This  deviation  from  exact  concurrence  is 
called  spherical  aberration. 

967.  Conjugate  Foci. — Let  P  (Fig.  674)  be  a  luminous  point  situ- 
ated on  the  principal  axis  of  a  spherical  mirror,  and  let  P I  be  one  of 
the  rays  which  it  sends  to  the  mirror.  Draw  the  normal  O  I,  which 
is  simply  a  radius  of  the  sphere.  Then  0 1 P  is  the  angle  of  incid- 


CONCAVE   MIRROR.  979 

ence,  and  the  angle  of  reflection  OIF  must  be  equal  to  it;  hence 
O  I  bisects  an  angle  of  the  triangle  P  I  P',  and  therefore  we  have 

IP   _  OP 
IP  ~  OF 

Let  p,p  denote  A  P,  A  P'  respectively,  and  let  r  denote  the  radius  of 
the  sphere.  Then,  if  the  angular  aperture  of  the  mirror  is  small,  I  P 
is  sensibly  equal  to  p,  and  I  P'  to  p'.  Substituting  these  approxi- 
mate values,  the  preceding  equation  becomea 


Pf  =   *—  ^j  whence  pr  +  p'  r  =  '2pp'; 


or,  dividing  by  pp'r, 


I,  =  2-.  («) 

p'        r 


This  formula  determines  the  position  of  the  point  P',  in  which  the 
reflected  ray  cuts  the  principal  axis,  and  shows  that  it  is,  to  the 
accuracy  of  our  approximation,  independent  of  the  position  of  the 
point  I;  that  is  to  say,  all  the  rays  which  P  sends  to  the  mirror  are 
reflected  to  the  same  point  P'.  We  have  assumed  P  to  be  on  the 
principal  axis.  If  we  had  taken  it  on  a  secondary  axis,  as  at  p  (Fig. 
674),  we  should  have  found,  by  the  same  process  of  reasoning,  that 
the  reflected  rays  would  all  meet  in  a  point  p  on  that  secondary 
axis.  The  distinction  between  primary  and  secondary  axes,  in  the 
case  of  a  spherical  mirror,  is  in  fact  merely  a  matter  of  convenience, 
not  representing  any  essential  difference  of  property.  Hence  we  can 
lay  down  the  following  general  proposition  as  true  within  limits  of 
error  corresponding  to  the  approximate  equalities  which  we  have 
above  assumed  as  exact:  — 

Rays  proceeding  from  any  given  point  in  front  of  a  concave 
spherical  mirror,  are  reflected  so  as  to  meet  in  another  point;  and 
the  line  joining  the  tivo  points  passes  through  the  centre  of  the  sphere. 

It  is  evident  that  rays  proceeding  from  the  second  point  to  the 
mirror,  would  be  reflected  to  the  first.  The  relation  between  'them 
is  therefore  mutual,  and  they  are  hence  called  conjugate  foci.  By  a 
focus  in  general  is  meant  a  point  in  which  a  number  of  rays,  which 
originally  came  from  the  same  point,  meet  (or  would  meet  if  pro- 
duced); and  the  rays  which  thus  meet,  taken  collectively,  are  called 
a  pencil.  Fig.  675  represents  two  pencils  of  rays  whose  foci  S  s  are 
conjugate,  so  that,  if  either  of  them  be  regarded  as  an  incident  pencil, 
the  other  will  be  the  corresponding  reflected  pencil. 


980  REFLECTION   OF   LIGHT. 

We  can  now  explain  the  formation  of  images  by  concave  mirrors. 
Each  point  of  the  object  sends  a  pencil  of  rays  to  the  mirror,  which 
converge,  after  reflection,  to  the  conjugate  focus.  If  the  eye  of  the 
observer  be  placed  beyond  this  point  of  concourse,  and  in  the  path 
of  the  rays,  they  will  present  to  him  the  same  appearance  as  if  they 


Fig.  675.—  Conjugate  Foci 

had  come  from  this  point  as  origin.     The  image  is  thus  composed  of 

points  which  are  the  conjugate  foci  of  the  several  points  of  the  object. 

968.  Principal  Focus.  —  If,  in  formula  (a)  of  last  section,  we  make 

p  jincrease  continually,  the  term  -  will  continually  decrease,  and  will 

vanish  as  p  becomes  infinite.  This  is  the  case  of  rays  parallel  to  the 
principal  axis,  for  parallel  rays  may  be  regarded  as  coming  from  a 
point  at  infinite  distance.  The  formula  then  becomes 


that  is  to  say,  the  principal  focal  distance  is  half  the  radius  of  cur- 
vature. This  distance  is  often  called  the  focal  length  of  the  mirror. 
If  we  denote  it  by  /,  the  general  formula  becomes 

i  +  M> 

969.  Discussion  of  the  Formula.  —  By  the  aid  of  this  formula  wo 
can  easily  trace  the  corresponding  movements  of  conjugate  foci. 

If  p  is  positive  and  very  large,  p'  is  a  very  little  greater  than  /; 
that  is  to  say,  the  conjugate  focus  is  a  very  little  beyond  the  princi- 
pal focus. 

As  p  diminishes,  p'  increases,  until  they  become  equal,  in  which 
case  each  of  them  is  equal  to  r  or  2/;  that  is  to  say,  the  conjugate 
foci  move  towards  each  other  till  they  coincide  at  the  centre  of  cur- 
vature. This  last  result  is  obvious  in  itself;  for  rays  from  the  centre 


CONCAVE   MIRROR.  981 

of  curvature  are  normal  to  the  mirror,  and  are  therefore  reflected 
directly  back. 

As  p  continues  to  diminish,  the  two  foci,  as  it  were,  change  places ; 
the  luminous  point  advancing  from  the  centre  of  curvature  to  the 
principal  focus,  while  the  conjugate  focus  moves  away  from  the  centre 
of  curvature  to  infinity. 

As  the  luminous  point  continues  to  approach  the  mirror,  -  is 

greater  than  *  and  hence  L,  and  therefore  also  p',  must  be  nega- 

/  p 

tive.  The  physical  interpretation  of  this  result  is  that  the  conjugate 
focus  is  behind  the  mirror,  as  at  s  (Fig.  676),  and  that  the  reflected 


Fig.  676.— Virtual  Focus. 

rays  diverge  as  if  they  had  come  from  this  point.  Such  a  focus  is 
called  virtual,  while  a  focus  in  which  rays  actually  meet  is  called 
real.  As  the  luminous  point  moves  up  from  F  to  the  mirror,  the 
conjugate  focus  moves  up  from  an  infinite  distance  at  the  back,  and 
meets  it  at  the  surface  of  the  mirror. 

If  S  is  a  real  luminous  point  sending  rays  to  the  mirror,  it  must 
of  necessity  lie  in  front  of  the  mirror,  and  p  therefore  cannot  be  nega- 
tive; but  when  we  are  considering  images  of  images  this  restriction 
no  longer  holds.  If  an  incident  beam,  for  example,  converges  to- 
wards a  point  s  at  the  back  of  the  mirror,  it  will  be  reflected  to  a 
point  S  in  front.  In  this  case  p  is  negative,  and  p'  positive.  The 
conjugate  foci  S  s  have  in  fact  changed  places. 

It  appears  from  the  above  investigation  that  there  are  two '  prin- 
cipal cases,  as  regards  the  positions  of  conjugate  foci  of  a  concave  mirror. 

1.  One  focus  between  F  and  C;  and  the  other  beyond  C. 

2.  One  focus  between  F  and  the  mirror;  and  the  other  behind 
the  mirror. 

In  the  former  case,  the  foci  move  to  meet  each  other  at  C;  in  the 
latter,  they  move  to  meet  each  other  at  the  surface  of  the  mirror. 


982  REFLECTION  OF  LIGHT. 

970.  Formation  of  Images. — We  are  now  in  a  position  to  discuss 
the  formation  of  images  by  concave  mirrors.  Let  AB  (Fig.  677)  be 
an  object  placed  in  front  of  a  concave  mirror,  at  a  distance  greater 
than  its  radius  of  curvature.  All  the  rays  which  diverge  from  A 
will  be  reflected  to  the  conjugate  focus  a.  Hence  this  point  can  be 
found  by  the  following  construction.  Draw  through  A  the  ray  AA' 
parallel  to  the  principal  axis,  and  draw  its  path  after  reflection,  which 
must  of  necessity  pass  through  the  principal  focus.  The  intersection 
of  this  reflected  ray  with  the  secondary  axis  through  A  will  be  the 
point  required.  A  similar  construction  will  give  the  conjugate  focus 


Fig.  677.— Formation  of  Image. 

corresponding  to  any  other  point  of  the  object;  b,  for  example,1  is 
the  focus  conjugate  to  B.  Points  of  the  object  lying  between  A  and 
B  will  have  their  conjugate  foci  between  a  and  6.  An  eye  placed 
behind  the  object  AB  will  accordingly  receive  the  same  impression 
from  the  reflected  rays  as  if  the  image  a  b  were  a  real  object. 

Since  the  lines  joining  corresponding  points  of  object  and  image 
cross  at  the  point  C,  which  lies  between  them  when  the  image  is 
real,  a  real  image  formed  by  a  concave  mirror  is  always  inverted. 

971.  Size  of  Image. — As  regards  the  comparative  sizes  of  object 
and  image,  it  is  obvious,  from  similar  triangles,  that  their  linear 
dimensions  are  directly  as  their  distances  from  C,  the  centre  of 
curvature. 

Again,  we  have  proved  in  §  9G7  that,  in  the  notation  of  that 
section, 

IZ  -  P_p- 
IF       OP" 

1  It  is  only  by  accident  that  6  happens  to  lie  on  A  A'  in  the  figure. 


SIZE    OF    IMAGE. 


983 


or,  the  distances  of  object  and  image  from  the  mirror  are  directly  as 
their  distances  from  the  centre  of  curvature.     Their  linear  dimen- 
sions are  therefore  directly  as  their  distances  from  the  mirror. 
Again,  by  equation  (6), 

1  _  i  _  i  _  P^f 
P'     f      p~  fp' 
whence 

?  •  "/• 

where  p—f  is  the  distance  of  the  object  from  the  principal  focus. 
Hence  the  linear  dimensions  of  object  and  image  are  in  the  ratio  of 
the  distance  of  the  object  from  the  principal  focus  to  the  focal  length. 

These  three  rules  are  perfectly  general,  both  for  concave  and 
convex  mirrors. 

The  first  rule  shows  that  the  object  and  image  are  equal  when 


Fig.  (578.— Experiment  of  Phantom  Bouquet. 


they  coincide  at  the  reflecting  surface,  and  that,  as  they  separate 
from  this  point  in  opposite  directions,  that  which  moves  away  from 
the  centre  of  curvature  continually  gains  in  size  upon  the  other. 
The  second  rule  shows  that  the  object  and  image  are  equal  when 


984 


REFLECTION   OF   LIGHT. 


they  coincide  at  the  centre  of  curvature,  and  that  as  they  separate 
from  this  point,  in  opposite  directions,  that  which  moves  away  from 
the  mirror  continually  gains  in  size  upon  the  other. 

The  third  rule  shows  that,  when  the  object  is  at  the  principal 
focus,  the  size  of  the  image  is  infinite. 

972.  Experiment  of  the  Phantom  Bouquet. — Let  a  box,  open  on  one 
side,  be  placed  in  front  of  a  concave  mirror  (Fig.  678),  at  a  distance 
about  equal  to  its  radius  of  curvature,  and  let  an  inverted  bouquet  be 
suspended  within  it,  the  open  side  of  the  box  being  next  the  mirror. 
By  giving  a  proper  inclination  to  the  mirror,  an  image  of  the  bouquet 
will  be  obtained  in  mid-air,  just  above  the  top  of  the  box.     As  the 
bouquet  is  inverted,  its  image  is  erect,  and  a  real  vase  may  be  placed 
in  such  a  position  that  the  phantom  bouquet  shall  appear  to  be  stand- 
ing in  it.    The  spectator  must  be  full  in  front  of  the  mirror,  and  at  a 
sufficient  distance  for  all  parts  of  the  image  to  lie  between  his  eyes 
and  the  mirror.     When  the  colours  of  the  bouquet  are  bright,  the 
image  is  generally  bright  enough  to  render  the  illusion  very  complete. 

973.  Images  on  a  Screen. — Such  experiments  as  that  just  described 


Fig.  679. — Image  on  Screen. 


can  only  be  seen  by  a  few  persons  at  once,  since  they  require  the 
spectator  to  be  in  a  line  with  the  image  and  the  mirror.  When  an 
image  is  projected  on  a  screen,  it  can  be  seen  by  a  whole  audience 


IMAGES   ON    SCREEN    AND    IN    MID   AIR. 


985 


at  once,  if  the  room  be  darkened  and  the  image  be  large  and  bright. 
Let  a  lighted  candle,  for  example,  be  placed  in  front  of  a  concave 
mirror,  at  a  distance  exceeding  the  focal  length,  and  let  a  screen  be 
placed  at  the  conjugate  focus;  an  inverted  image  of  the  candle  will 
be  depicted  on  the  screen.  Fig.  679  represents  the  case  in  which  the 
candle  is  at  a  distance  less  than  the  radius  of  curvature,  and  the 
image  is  accordingly  magnified. 

By  this  mode  of  operating,  the  formula  for  conjugate  focal  dis- 
tances can  be  experimentally  verified  with  considerable  rigour,  care 
being  taken,  in  each  experiment,  to  place  the  screen  in  the  position 
which  gives  the  most  sharply  defined  image. 

974.  Difference  between  Image  on  Screen,  and  Image  as  seen  in 
Mid-air.  Caustics. — For  the  sake  of  simplicity  we  have  made  some 
statements  regarding  visible  images  which  are  not  quite  accurate; 
and  we  must  now  indicate  the  necessary  corrections. 

Images  thrown  on  a  screen  have  a  determinate  position,  and  are 
really  the  loci  of  the  conjugate  foci  of  the  points  of  the  object;  but 


_ *-.-.- 


Fig.  680.— Position  of  Image  in  Oblique  Reflection. 

this  is  not  rigorously  true  of  images  seen  directly.    They  change  their 
position  to  some  extent,  according  to  the  position  of  the  observer. 

The  actual  state  of  things  is  explained  by  Fig.  680.  The  plane 
of  the  figure1  is  a  principal  plane  (that  is,  a  plane  containing  the  prin- 
cipal axis)  of  a  concave  hemispherical  mirror,  and  the  incident  rays 

1  Figs.  680  and  698  are  borrowed,  by  permission,  from  Mr.  Osmund  Airy's  Geometrical 
Optic$. 


98G  REFLECTION   OF   LIGHT. 

are  parallel  to  the  principal  axis.  All  the  rays  reflected  in  the  plane 
of  the  figure  touch  a  certain  curve  called  a  caustic  curve,  which  has 
a  cusp  at  F,  the  principal  focus;  and  the  direction  in  which  the 
image  is  seen  by  an  eye  situated  in  the  plane  of  the  figure  is  deter- 
mined by  drawing  from  the  eye  a  tangent  to  this  caustic.  If  the 
eye  be  at  E,  on  the  principal  axis,  the  point  of  contact  will  be  F; 
but  when  the  rays  are  received  obliquely,  as  at  E',  it  will  be  at  a 
point  a  not  lying  in  the  direction  of  F.  For  an  eye  thus  situated,  a 
is  called  the  primary  focus,  and  the  point  where  the  tangent  at  a 
cuts  the  principal  axis  is  called  the  secondary  focus.  When  the  eye 
is  moved  in  the  plane  of  the  diagram,  the  apparent  position  of  the 
image  (as  determined  by  its  remaining  in  coincidence  with  a  cross  of 
threads  or  other  mark)  is  the  primary  focus;  and  when  the  eye  is 
moved  perpendicular  to  the  plane  of  the  diagram,  the  apparent  posi- 
tion of  the  image  is  the  secondary  focus.1  If  we  suppose  the  diagram 
to  rotate  about  the  principal  axis,  it  will  still  remain  true  in  all 
positions,  and  the  surface  generated  by  this  revolution  of  the  caustic 
curve  is  the  caustic  surface.  Its  form  and  position  vary  with  the 
position  of  the  point  from  which  the  incident  rays  proceed;  and  it 
has  a  cusp  at  the  focus  conjugate  to  this  point. 

There  is  alwavs  more  or  less  blurring,  in  the  case  of  images  seen 
obliquely  (except  in  plane  mirrors),  by  reason  of  the  fact  that  the 
point  of  contact  with  the  caustic  surface  is  not  the  same  for  rays 

entering   different   parts   of   the 
pupil  of  the  eye. 

A  caustic  curve  can  be  exhibited 
experimentally  by  allowing  the 
rays  of  the  sun  or  of  a  lamp  to 
fall  on  the  concave  surface  of  a 
strip  of  polished  metal  bent  into 
the  form  of  a  circular  arc,  as  in 
Fig.  681,  the  reflected  light  being 
received  on  a  sheet  of  white  paper 
on  which  the  strip  rests.  The 

Fig.  esi.-caustic  by  Reflection.  same  effect  may  often  be  observed 

on  the  surface  of  a  cup  of  tea,  the 
reflector  in  this  case  being  the  inside  of  the  tea-cup. 

1  Since  every  ray  incident  parallel  to  the  principal  axis,  is  reflected  through  the  principal 
axis.  If  the  incident  rays  diverged  from  a  point  on  the  principal  axis,  they  would  still  be 
reflected  through  the  principal  axis. 


FOCAL  LINES. 


987 


The  image  of  a  luminous  point  received  upon  a  screen  is  formed 
by  all  the  rays  which  touch  the  corresponding  caustic  surface.  The 
brightest  and  most  distinct  image  will  be  formed  at  the  cusp,  which 
is,  in  fact,  the  conjugate  focus;  but  there  will  be  a  border  of  fainter 
light  surrounding  it.  This  source  of  indistinctness  in  images  is  an 
example  of  spherical  aberration  (§  967). 

975.  Image  on  a  Screen  by  Oblique  Reflection. — If  we  attempt  to 
throw  upon  a  screen  the  image  of  a  luminous  point  by  means  of  a 
concave  mirror  very  oblique  to  the  incident  rays,  we  shall  find  that 
no  image  can  be  obtained  at  all  resembling  a  point;  but  that  there 
are  two  positions  of  the  screen  in  which  the  image  becomes  a 
line. 

In  the  annexed  figure  (Fig.  682),  which  represents  on  a  larger 
scale  a  portion  of  Fig.  680,  ac,bd  are  rays  from  the  highest  and 
lowest  points  of  the  portion 
R  S  of  the  hemispherical  mir- 
ror, which  portion  we  suppose 
to  be  small  in  both  its  dimen- 
sions in  comparison  with  the 
radius  of  curvature;  and  we 
may  suppose  the  rest  of  the 
hemisphere  to  be  removed,  so 
that  R  S  will  represent  a  small 
concave  mirror  receiving  a 
pencil  very  obliquely. 


Then,  if  a  screen  be  held 
perpendicular  to  the  plane  of 
the  diagram,  at  m,  where  the 
section  of  the  pencil  by  the  plane  of  the  diagram  is  narrowest,  a 
blurred  line  of  light  will  be  formed  upon  it,  the  length  of  the  line 
being  perpendicular  to  the  plane  of  the  diagram.  This  is  called  the 
primary  focal  line. 

The  secondary  focal  line  is  c  d,  which,  if  produced,  passes  through 
the  centre  of  curvature  of  the  mirror,  and  also  through  the  point 
from  which  the  incident  light  proceeds.  This  line  is  very  sharply 
formed  upon  a  screen  held  so  as  to  coincide  with  c  d  and  to  be  per- 
pendicular to  the  plane  of  the  diagram.  Its  edges  are  much  better 
defined  than  those  of  the  primary  line;  and  its  position  in  space  is 
also  more  definite.  If  the  mirror  is  used  as  a  burning-glass  to  collect 


Fig.  6S2. -Formation  of  Focal  Lines. 


988  REFLECTION   OF  LIGHT. 

the  sun's  rays,  ignition  will  be  more  easily  obtained  at  one  of  these 
lines  than  in  any  intermediate  position. 

Focal  lines  can  also  be  seen  directly.  In  this  case  a  small  element 
of  the  mirror  sends  all  its  reflected  rays  to  the  eye,  the  rays  from 
opposite  sides  of  the  element  crossing  each  other  at  the  focal  lines, 
before  they  reach  the  eye.  It  is  possible,  in  certain  positions  of  the 
eye,  to  see  either  focal  line  at  pleasure,  by  altering  the  focal  adjust- 
ment of  the  eye;  or  the  two  may  be  seen  with  imperfect  definition 
crossing  each  other  at  right  angles.  The  experiment  is  easily  made 
by  employing  a  gas  flame,  turned  very  low,  as  the  source  of  light.  One 
line  is  in  the  plane  of  incidence,  and  the  other  is  normal  to  this  plane. 

976.  Virtual  Image  in  Concave  Mirror. — Let  an  object  be  placed,  as 
in  Figs.  683, 684,  in  front  of  a  concave  mirror,  at  a  distance  less  than 

that  of  the  principal  focus. 
The  rays  incident  on  the 
mirror  from  any  point  of  it, 
as  A  (Fig.  683),  will  be  re- 
flected as  a  divergent  pencil, 
the  focus  from  which  they 
diverge  being  a  point  b  at 
the  back  of  the  mirror.  To 
find  this  point,  we  may  trace 
the  course  of  a  ray  through 
A  parallel  to  the  principal 

3.-Formation  of  Virtual  Image.  axis.      Such    a    ray   will    be 

reflected    to   the   principal 

focus  F,  and  by  producing  this  reflected  ray  backwards  till  it 
meets  the  secondary  axis  C  A,  the  point  6,  which  is  the  conjugate 
focus  of  A,  is  determined.  We  can  find  in  the  same  way  the  posi- 
tion of  a,  the  conjugate  focus  of  B,  and  it  is  obvious  that  the  image 
of  A  B  will  be  erect  and  magnified. 

977.  Remarks  on  Virtual  Images. — A  virtual  image  cannot  be  pro- 
jected on  a  screen;  for  the  rays  which  produce  it  do  not  actually 
pass  through  its  place,  but  only  seem  to  do  so.     A  screen  placed  at 
a  b  would  obviously  receive  none  of  the  reflected  light  whatever. 

1  The  "elongated  figure  of  8"  which  is  often  mentioned  in  connection  with  the  secondary 
focal  line,  is  obtained  by  turning  the  screen  about  n  the  middle  point  of  c  d,  so  as  to  blur 
both  ends  of  the  image  by  bad  focussing.  It  will  be  observed,  from  an  inspection  of  tho 
diagram,  that  cdia  very  oblique  to  the  reflected  rays. 

If  we  neglect  the  blurring  of  the  primary  line,  we  may  describe  the  part  of  the  pencil 
lying  between  the  two  lines  as  a  tetrahedron,  of  which  the  two  lines  are  opposite  edges. 


CONVEX   MIRROR. 


989 


The  images  seen  in  a  plane  mirror  are  virtual;  and  any  spherical 
mirror,  whether  concave  or  convex,  is  nearly  equivalent  to  a  plane 
mirror,  when  the  distance  of  the  object  from  its  surface  is  small  in 
comparison  with  the  radius 
of  curvature. 

978.  Convex  Mirrors.  —  It 
is  easily  shown,  by  a  simple 
construction,  that  rays  inci- 
dent from  any  luminous 
point  upon  a  convex  mirror, 
diverge  after  reflection.  The 
principal  focus,  and  the  foci 
conjugate  to  all  points  ex- 
ternal to  the  sphere,  are 
therefore  virtual. 

To  adapt  formulae  (a)  and 
(b)  of  the  preceding  sections 
to  the  case  of  convex  mirrors, 
we  have  only  to  alter  the 

sign  of  the  term    —  or   j- 

so  that  for  a  convex  mirror 
we  shall  have 


Fig.  684.— Virtual  Image  In  Concave  Mirror. 


r  and  /  being  here  regarded  as  essentially  positive. 

From  this  formula  it  is  obvious  that  one  at  least  of  the  two  dis- 
tances p,p  must  be  negative;  that  is  to  say,  one  at  least  of  any  pair 
of  conjugate  foci  must  lie  behind  the  mirror. 

The  construction  for  an  image  (Fig.  685)  is  the  same  as  in  the 
case  of  concave  mirrors.  Through  any  selected  point  of  the  object 
draw  a  ray  parallel  to  the  principal  axis;  the  reflected  ray,  if  pro- 
duced backwards,  must  pass  through  the  principal  focus,  and  its 
intersection  with  the  secondary  axis  through  the  selected  point  deter- 
mines the  corresponding  point  of  the  image.  The  image  of  an  external 
object  will  evidently  be  erect,  and  smaller  than  the  object.  Repeat- 
ing the  same  construction  when  the  object  is  nearer  to  the  mirror,  we 
see  that  the  image  will  be  larger  than  before. 

The  linear  dimensions  of  an  object  and  its  image,  whether  in  the 
case  of  a  convex  or  a  concave  mirror,  are  directly  proportional  to 


990  REFLECTION   OF  LIGHT. 

their  distances  from  the  centre  of  curvature,  and  are  also  directly 
proportional  to  their  distances  from  the  mirror.  The  image  is  in- 
verted or  erect  according  as  the  centre  of  curvature  does  or  does  not 


Fig.  686.— Formation  of  Image  in  Convex  Mirror. 

lie  between  the  object  and  its  image.  In  the  case  of  a  convex 
mirror  the  centre  never  lies  between  them  (if  the  object  be  real), 
and  therefore  the  image  is  always  erect. 

Convex  mirrors  are  very  seldom  employed  in  optical  instru- 
ments. 

The  silvered  globes  which  are  frequently  used  as  ornaments,  are 
examples  of  convex  mirrors,  and  present  to  the  observer  at  one  view 
an  image  of  nearly  the  whole  surrounding  landscape.  As  the  part 
of  the  mirror  in  which  he  sees  this  image  is  nearly  an  entire  hemi- 
sphere, the  deformation  of  the  image  is  very  notable,  straight  lines 
being  reflected  as  curves. 

979.  Anamorphosis. — Much  greater  deformations  are  produced  by 
cylindric  mirrors.    A  cylindric  mirror,  when  the  axis  of  the  cylinder 
is  vertical,  behaves  like  a  plane  mirror  as  regards  the  angular  magni- 
tude under  which  the  height  of  the  image  is  seen,  and  like  a  spherical 
mirror  as  regards  the  breadth  of  the  image.      If  it  be  a  convex  cylin- 
der, it  causes  bodies  to  appear  unduly  contracted  horizontally  in  pro- 
portion to  their  heights.     Distorted  pictures  are  sometimes  drawn 
upon  paper,  according  to  such  a  system  that  when  they  are  seen 
reflected  in  a  cylindric  mirror  properly  placed,  as  in  Fig.  686,  the 
distortion  is  corrected,  and  while  the  picture  appears  a  mass  of  con- 
fusion, the  image  is  instantly  recognized.     This  restoration  of  true 
proportion  in  a  picture  is  called  anamorphosis. 

980,  Medical  Applications. — Concave  mirrors  are  frequently  used 


OPHTHALMOSCOPE   AND   LARYNGOSCOPE. 


991 


t)  concentrate  light  upon  an  object  for  the  purpose  of  rendering  it 
more  distinctly  visible. 

The  ophthalmoscope  is  a  small  concave  mirror,  with  a  small  hole  in 
its  centre,  through  which  the  observer  looks  from  behind,  while  he 


Fig.  686. — Anamorphosis. 

directs  a  beam  of  reflected  light  from  a  lamp  into  the  pupil  of  the 
patient's  eye.  In  this  way  (with  the  help  sometimes  of  a  lens)  the 
retina  can  be  rendered  visible,  and  can  be  minutely  examined. 

The  laryngoscope  consists  of  two  mirrors.  One  is  a  small  plane 
mirror,  with  a  handle  attached,  at  an  angle  of  about  45°  to  its  plane. 
This  small  mirror  is  held  at  the  back  of  the  patient's  mouth,  so  that 
the  observer,  looking  into  it,  is  able  by  reflection  to  see  down  the 
patient's  throat,  the  necessary  illumination  being  supplied  by  a  con- 
cave mirror,  strapped  to  the  observer's  forehead,  by  means  of  which 
the  light  from  a  lamp  is  reflected  upon  the  plane  mirror,  which  again 
reflects  it  down  the  throat. 

Some  additions  to  this  chapter  will  be  found  at  page  1086. 


CHAPTER    LXIX. 


REFRACTION. 


981.  Refraction. — When  a  ray  of  light  passes  from  one  transparent 
medium  to  another,  it  undergoes  a  change  of  direction  at  the  surface 

of  separation,  so  that  its  course  in 
the  second  medium  makes  an  angle 
with  its  course  in  the  first.  This 
changing  of  direction  is  called  re- 
fraction. 

The  phenomenon  can  be  exhibited 
by  admitting  a  beam  of  the  sun's 
rays  into  a  dark  room,  and  receiv- 
ing it  on  the  surface  of  water  con- 
tained in  a  rectangular  glass  vessel 
(Fig.  687).  The  path  of  the  beam 
will  be  easily  traced  by  its  illumi- 
nation of  the  small  solid  particles 
which  lie  in  its  course. 

The  following  experiment  is  a 
well-known  illustration  of  refrac- 
tion:— A  coin  m  n  (Fig.  688)  is 
laid  at  the  bottom  of  a  vessel  with 

opaque  sides,  and  a  spectator  places  himself  so  that  the  coin  is  just 
hidden  from  him  by  the  side  of  the  vessel;  that  is  to  say,  so  that  the 
line  m  A  in  the  figure  passes  just  above  his  eye.  Let  water  now  be 
poured  into  the  vessel,  care  being  taken  not  to  displace  the  coin. 
The  bottom  of  the  vessel  will  appear  to  rise,  and  the  coin  will  come 
into  sight.  Hence  a  pencil  of  rays  from  m  must  have  entered  the 
spectator's  eye.  The  pencil  in  fact  undergoes  a  sudden  bend  at  the 
surface  of  the  water,  and  thus  reaches  the  eye  by  a  crooked  course., 


Fig.  687.—  Refractior 


LAWS   OF   REFRACTION. 


993 


—Experiment  of  Coin  in  Basin, 


in  which  the  obstacle  A  is  evaded.     If  the  part  of  the  pencil  in  air 

be  produced  backwards,  its  rays  will  approximately  meet  in  a  point 

m',  which  is  therefore   the 

image  of  m.     Its  position  is 

not  correctly  indicated  in  the 

figure,  being  placed  too  much 

to  the  left  (§  990). 

The  broken  appearance 
presented  by  a  stick  (Fig. 
689)  when  partly  immersed 
in  water  in  an  oblique 
position,  is  similarly  ex- 
plained, the  part  beneath  the  water  being  lifted  up  by  refraction. 

982.  Refractive  Powers  of  Different  Media. — In  the  experiments  of 
the  coin  and  stick,  the  rays,  in  leaving  the  water,  are  bent  away 
from  the  normals  Z I N,  Z'  I'  N' 

at  the  points  of  emergence;  in 
the  experiment  first  described 
(Fig.  687),  on  the  other  hand, 
the  rays,  in  passing  from  air  into 
water,  are  bent  nearer  to  the 
normal.  In  every  case  the  path 
which  the  rays  pursue  in  going 
is  the  same  as  they  would  pur- 
sue in  returning;  and  of  the 
two  media  concerned,  that  in 
which  the  ray  makes  the  smaller 
angle  with  the  normal  is  said  to 

have  greater  refractive  power  than  the  other,  or  to  be  more  highly 
refracting. 

Liquids  have  greater  refractive  power  than  gases,  and  as  a  general 
rule  (subject  to  some  exceptions  in  the  comparison  of  dissimilar  sub- 
stances) the  denser  of  two  substances  has  the  greater  refracting  power. 
Hence  it  has  become  customary,  in  enunciating  some  of  the  laws  of 
optics,  to  speak  of  the  denser  medium  and  the  rarer  medium,  when  the 
more  correct  designations  would  be  more  refractive  and  less  refractive. 

983.  Laws  of  Refraction. — The  quantitative  law  of  refraction  was 
not  discovered  till  quite  modern  times.    It  was  first  stated  by  Snell, 
a  Dutch  philosopher,  and  was  made  more  generally  known  by  Des- 
cartes, who  has  often  been  called  its  discoverer. 


Fig.  689.— Appearance  of  Stick  in  Water. 


994  REFRACTION. 

Let  R I  (Fig.  690)  be  a  ray  incident  at  I  on  the  surface  of  separa- 
tion of  two  media,  and  let  I  S  be  the  course  of  the  ray  after  refrac- 
tion. Then  the  angles  which  R I  and  I  S  make  with  the  normal 
are  called  the  angle  of  incidence  and  the  angle  of  refraction  respec- 
tively; and  the  first  law  of  refraction 
is  that  these  angles  lie  in  the  same 
plane,  or  the  plane  of  refraction  is 
the  same  as  the  plane  of  incidence. 

The  law  which  connects  the  mag- 
nitudes of  these  two  angles,  and  which 
was  discovered  by  Snell,  can  only  be 
stated  either  by  reference  to  a  geo- 
metrical construction,  or  by  employ- 
ing the  language  of  trigonometry. 
Describe  a  circle  about  the  point  of 
incidence  I  as  centre,  and  drop  per- 
Fig.  690.— Law  of  Refraction.  pendiculars,  from  the  points  where 

it  cuts  the  rays,  on  the  normal.    The 

law  is  that  these  perpendiculars  R/  P',  S  P,  will  have  a  constant  ratio; 
or  the  sines  of  the  angles  of  incidence  and  refraction  are  in  a  con- 
stant ratio.  It  is  often  referred  to  as  the  law  of  sines. 

The  angle  by  which  a  ray  is  turned  out  of  its  original  course  in 
undergoing  refraction  is  called  its  deviation.  It  is  zero  if  the  in- 
cident ray  is  normal,  and  always  increases  with  the  angle  of  inci- 
dence. 

984.  Verification  of  the  Law  of  Sines. — These  laws  can  be  verified 
by  means  of  the  apparatus  represented  in  Fig.  691,  which  is  very 
similar  to  that  employed  by  Descartes.  It  has  a  vertical  divided 
circle,  to  the  front  of  which  is  attached  a  cylindrical  vessel,  half -filled 
with  water  or  some  other  transparent  liquid.  The  surface  of  the 
liquid  must  pass  exactly  through  the  centre  of  the  circle.  I  is  a 
movable  mirror  for  directing  a  reflected  beam  of  solar  light  on  the 
centre  O.  The  beam  must  be  directed  centrally  through  a  short 
tube  attached  to  the  mirror,  and  to  facilitate  this  adjustment  the 
tube  is  furnished  with  a  diaphragm  with  a  hole  in  its  centre.  The 
arm  O  a  is  movable  about  the  centre  of  the  circle,  and  carries  a  ver- 
nier for  measuring  the  angle  of  incidence.  The  ray  undergoes  refrac- 
tion at  0;  and  the  angle  of  refraction  is  measured  by  means  of  a 
second  arm  O  R,  which  is  to  be  moved  into  such  a  position  that  the 
diaphragm  of  its  tube  receives  the  beam  centrally.  No  refraction 


LAW   OF   SINES. 


995 


occurs  at  emergence,  since  the  emergent  beam  is  normal  to  the  sur- 
faces of  the  liquid  and 
glass;  the  position  of 
the  arm  accordingly 
indicates  the  direction 
of  the  refracted  ray. 
The  angles  of  inci- 
dence and  refraction 
can  be  read  off  at  the 
verniers  carried  by 
the  two  arms;  and  the 
ratio  of  their  sines 
will  be  found  con- 
stant. The  sines  can 
also  be  directly  mea- 
sured by  employing 
sliding-seales  as  indi- 
cated in  the  figure, 
the  readings  being 
taken  at  the  extre- 
mity of  each  arm. 

It  would  be  easy 

to  make  a  beam  of  light  enter  at  the  lower  side  of  the  apparatus,  in 
a  radial  direction;  and  it  would  be 
found  that  the  ratio  of  the  sines  was 
precisely  the  same  as  when  the  light 
entered  from  above.  This  is  merely 
an  instance  of  the  general  law,  that 
the  course  of  a  returning  ray  is  the 
same  as  that  of  a  direct  ray. 

985.  Airy's  Apparatus.— The  fol- 
lowing apparatus  for  the  same  pur- 
pose was  invented,  many  years  ago, 
by  the  present  astronomer  royal. 
B'  is  a  slider  travelling  up  and  down 
a  vertical  stem.  A  C'  and  B  C  are 
two  rods  pivoted  on  a  fixed  point 
B  of  the  vertical  stem.  C'  B'  and 
C  B'  are  two  other  rods  jointed  to 
the  former  at  C'  and  C,  and  pivoted  at  their  lower  ends  on  the  centre 


Fig.  691.— Apparatus  for  Verifying  the  Law. 


Fig.  692.— Airy '8  Apparatus. 


990  REFRACTION. 

of  the  slider.  B  C  is  equal  to  B'  C',  and  B  C'  to  B'  C.  Hence  the 
two  triangles  B  C  B',  B'  C'  B  are  equal  to  one  another  in  all  posi- 
tions of  the  slider,  their  common  side  B  B'  being  variable,  while  the 
other  two  sides  of  each  remain  unchanged  in  length  though  altered 
in  position. 

T>  r*          TV  fir 

The  ratio  CB/orc/B  is  made  equal  to  the  index  of  refraction  of  the 
liquid  in  which  the  observation  is  to  be  made.  For  water  this  ratio 
will  be  g.  Then,  if  the  apparatus  is  surrounded  with  water  up  to  the 

level  of  B,  A  B  C  will  be  the  path  of  a  ray,  and  a  stud  at  C  will 
appear  in  the  same  line  with  studs  at  A  and  B;  for  we  have 

sin  C'BB'  sin  C'  B  B'  C'B'  4 

lin  C  B  W       '     iaTC' B' B        =     C'B      :"     3* 

986.  Indices  of  Refraction. — The  ratio  of  the  sine  of  the  angle  of 
incidence  to  the  sine  of  the  angle  of  refraction  when  a  ray  passes 
from  one  medium  into  another,  is  called  the  relative  index  of  refrac- 
tion from  the  former  medium  to  the  latter.  When  a  ray  passes 
from  vacuum  into  any  medium  this  ratio  is  always  greater  than 
unity,  and  is  called  the  absolute  index  of  refraction,  or  simply  the 
index  of  refraction,  for  the  medium  in  question.  The  relative 
index  of  refraction  from  any  medium  A  into  another  B  is  always 
equal  to  the  absolute  index  of  B  divided  by  the  absolute  index  of  A. 
The  absolute  index  of  air  is  so  small  that  it  may  usually  be  neglected 
in  comparison  with  those  of  solids  and  liquids:  but  strictly  speaking, 
the  relative  index  for  a  ray  passing  from  air  into  a  given  substance 
must  be  multiplied  by  the  absolute  index  for  air,  in  order  to  obtain 
the  absolute  index  of  refraction  for  the  substance. 

The  following  table  gives  the  indices  of  refraction  of  several  sub- 
stances:— 

INDICES  OF  REFRACTION.* 


Diamond 2'44  to  2755 

Sapphire, 1794 

Flint-glass, T576  to  1'642 

Crown-glass, 1-581  to  1'563 

Rock-salt 1-545 

Canada  balsam, 1'540 

Bisulphide  of  carbon,  ....  1'678 
Linseed  oil  (sp.  gr.  -932),  .  .  .  1-482 
Oil  of  turpentine  (sp.  gr.  '885),  .  T478 


Alcohol, 1-372 

Aqueous  humour  of  eye,    ....  1*337 

Vitreous  humour, T339 

Crystalline  lens,  outer  coat,    .     .     .  1'337 

„             „      under  coat,  .     .     .  T379 

„             „      central  portion,     .  1'400 

Sea  water, 1'343 

Pure  water 1'336 

Air  at  0°  C.  and  760  *•....    1-000294 


987.  Critical  Angle. — We  see,  from  the  law  of  sines,  that  when  the 

1  The  index  of  refraction  is  always  greater  for  violet  than  for  red  (see  Chap.  Ixxii.).   The 
numbers  in  this  table  are  to  be  understood  as  mean  values. 


CRITICAL  ANGLE.  997 

incident  ray  is  in  the  less  refractive  of  the  two  media,  to  every  pos- 
sible angle  of  incidence  there  is  a  corresponding  angle  of  refraction. 
This,  however,  is  not  the  case  when  the  incident  ray  is  in  the  more 
refractive  of  the  two  media.  Let  S  0,  S'  O,  S"  O  (Fig.  693)  be  inci- 


Fig.  693.— Critical  Angle. 

dent  rays  in  the  less  refractive  medium,  and  0  R,  0  R',  0  R"  the 
corresponding  refracted  rays.  There  will  be  a  particular  direction 
of  refraction  O  L  corresponding  to  the  angle  of  incidence  of  90°. 
Conversely,  incident  rays  R  0,  R'  O,  R"  O,  in  the  more  refractive 
medium,  will  emerge  in  the  directions  O  S,  O  S',  O  S",  and  the  direc- 
tion of  emergence  for  the  incident  ray  L  0  will  be  0  B,  which  is 
coincident  with  the  bounding  surface. 

The  angle  L  O  N  is  called  the  critical  angle,  and  is  easily  computed 
when  the  relative  index  of  refraction  is  given.  For  let  f*  denote  this 
index  (the  incident  ray  being  supposed  to  be  in  the  less  refractive 
medium),  then  we  are  to  have 

sin  90°  1 

— ; =  u.  whence  sin  x  =  — J 

sin  x  fj. 

that  is,  the  sine  of  the  optical  angle  is  the  reciprocal  of  the  in^dex  of 
refraction. 

When  the  media  are  air  and  water,  this  angle  is  about  48°  30'. 
For  air  and  different  kinds  of  glass  its  value  ranges  from  38°  to  41°. 

If  a  ray,  as  I  O,  is  incident  in  the  more  refractive  medium,  at  an 
angle  greater  than  the  critical  angle,  the  law  of  sines  becomes  nuga- 
tory, and  experiment  shows  that  such  a  ray  undergoes  internal  re- 
flection in  the  direction  0  I',  the  angle  of  reflection  being  equal  to 


998 


REFRACTION. 


the  angle  of  incidence.  Reflection  occurring  in  these  circumstances 
is  nearly  perfect,  and  has  received  the  name  of  total  reflection.  Total 
reflection  occurs  when  rays  are  incident  in  the  more  refractive 
medium  at  an  angle  greater  than  ike  critical  angle. 

The  phenomenon  of  total  reflection  may  be  observed  in  several 
familiar  instances.  For  example,  if  a  glass  of  water,  with  a  spoon 
in  it  (Fig.  694),  is  held  above  the  level  of  the  eye,  the  under  side  of 


Fig.  694.— Total  Reflection. 

the  surface  of  the  water  is  seen  to  shine  like  a  brilliant  mirror,  and 
the  lower  part  of  the  spoon  is  seen  reflected  in  it.  Beautiful  effects 
of  the  same  kind  may  be  observed  in  aquariums. 


CAMEEA   LUCIDA.  999 

938.  Camera  Lucida. — The  camera  lucida  is  an  instrument  some- 
times employed  to  facilitate  the  sketching  of  objects  from  nature. 
It  acts  by  total  reflection,  and  may  have  various  forms,  of  which 
that  proposed  by  Wollaston,  and  represented  in  Figs.  695,  696,  is 


Fig.  095.— Section  of  Prism.  Fig.  696.  — Camera  Lucida. 

one  of  the  commonest.  The  essential  part  is  a  totally-reflecting 
prism  with  four  angles,  one  of  which  is  90°,  the  opposite  one  135°, 
and  the  other  two  each  67°  30'.  One  of  the  two  faces  which  contain 
the  right  angle  is  turned  towards  the  objects  to  be  sketched.  Rays 
incident  normally  on  this  face,  as  x  r,  make  an  angle  greatly  exceed- 
ing the  critical  angle  with  the  face  c  d,  and  are  totally  reflected  from 
it  to  the  next  face  d  a,  whence  they  are  again  totally  reflected  to  the 
fourth  face,  from  which  they  emerge  normally.1  An  eye  placed  so 
as  to  receive  the  emergent  rays  will  see  a  virtual  image  in  a  direc- 
tion at  right  angles  to  that  in  which  the  object  lies.  In  practice,  the 
eye  is  held  over  the  angle  a  of  the  prism,  in  such  a  position  that  one- 
half  of  the  pupil  receives  these  reflected  rays,  while  the  other  half 
receives  light  in  a  parallel  direction  outside  the  prism.  The  observer 
thus  sees  the  reflected  image  projected  on  a  real  back-ground,  which 
consists  of  a  sheet  of  paper  for  sketching.  He  is  thus  enabled  to  pass 
a  pencil  over  the  outlines  of  the  image;  pencil,  image,  and  paper  being 
simultaneously  visible.  It  is  very  desirable  that  the  image  should  lie 
in  the  plane  of  the  paper,  not  only  because  the  pencil  point  and  the 
image  will  then  be  seen  with  the  same  focussing  of  the  eye,  but  also 
because  parallax  is  .thus  obviated,  so  that  when  the  observer  shifts 
his  eye  the  pencil  point  is  not  displaced  on  the  image.  A  concave 
lens,  with  a  focal  length  of  something  less  than  a  foot,  is  therefore 

1  The  use  of  having  two  reflections  is  to  obtain  an  erect  image.     An  image  obtained  by 
one  reflection  would  be  upside  down. 


1000 


REFRACTION. 


placed  close  in  front  of  the  prism,  in  drawing  distant  objects.  By 
raising  or  lowering  the  prism  in  its  stand  (Fig.  696),  the  image  of 
the  object  to  be  sketched  may  be  made  to  coincide  with  the  plane  of 
the  paper. 

The  prism  is  mounted  in  such  a  way  that  it  can  be  rotated  either 
about  a  horizontal  or  a  vertical  axis;  and 
its  top  is  usually  covered  with  a  movable 
plate  of  blackened  metal,  having  a  semi- 
circular notch  at  one  edge,  for  the  observer 
to  look  through. 

989.  Images  by  Refraction  at  a  Plane 
Surface.— Let  0  (Fig.  697)  be  a  small  ob- 
ject in  the  interior  of  a  solid  or  liquid 
bounded  by  a  plane  surface  AB.  Let 
O  B  C  be  the  path  of  a  nearly  normal  ray, 
and  let  B  C  (the  portion  in  air)  be  pro- 

__  duced  backwards  to  meet  the  normal  in  I. 

rig.  697.-image  by  Refraction.  Then,  since  A I B  and  A  0  B  are  the  in- 
clinations of  the  two  portions  of  the  ray 

to  the  normal,  we  have  (if  p.  be  the  index  of  refraction  from  air  into 
the  substance) — 


sin  AIB 

sin  AOB 


OB 
IB' 


But  0  B  is  ultimately  equal  to  O  A,  and  I B  to  I  A.  Hence,  if  we 
make  A I  equal  to  — ,  all  the  emergent  rays  of  a  small  and  nearly 

normal  pencil  emitted  by  0  will,  if  produced  backwards,  intersect 
0  A  at  points  indefinitely  near  to  the  point  I  thus  determined.  If 
the  eye  of  an  observer  be  situated  on  the  production  of  the  normal 
OA,  the  rays  by  which  he  sees  the  object  0  constitute  such  a 
pencil.  He  accordingly  sees  the  image  at  I.  As  the  value  of  p  is 

g  for  water,  and  about  ^  for  glass,  it  follows  that  the  apparent  depth 
of  a  pool  of  clear  water  when  viewed  vertically  is  -^  of  the  true 
depth,  and  that  the  apparent  thickness  of  a  piece  of  plate-glass 
when  viewed  normally  is  only  |  of  the  true  thickness. 

990.— When  the  incident  pencil  (Fig.  698)  is  not  small,  but  includes 
rays  of  all  obliquities,  those  of  them  which  make  angles  with  the 
normal  less  than  the  critical  angle  N  Q  R  will  emerge  into  air;  and 
the  emergent  rays,  if  produced  backwards,  will  all  touch  a  certain 


APPARENT  PLACE   OF   IMAGE. 


1001 


caustic  surface,  which  has  the  normal  Q  N  for  its  axis  of  revolution, 
and  touches  the  surface  at  all  points  of  a  circle  of  which  N  R  is  the 


Fig.  60S. —Caustic  by  Eefractioa 

radius.  Wherever  the  eye  may  be  situated,  a  tangent  drawn  from 
it  to  the  caustic  will  be  the  direction  of  the  visible  image.  If  the 
observer  sees  the  image  with  both  eyes,  both  being  equidistant  from 
the  surface  and  also  equidistant  from  the  normal,  the  two  lines  of 
sight  thus  determined  (one  for  each  eye)  will  meet  at  a  point  on  the 
normal,  which  will  accordingly  be  the  apparent  position  of  the  image. 
If,  on  the  other  hand,  both  eyes  are  in  the  same  plane  containing  the 
normal,  the  two  lines  of  sight  will  intersect  at  a  point  between  the 
normal  and  the  observer. 

The  image,  whether  seen  with  one  eye  or  two,  approaches  nearer 
to  the  surface  as  the  direction  of  vision  becomes  more  oblique,  and 
ultimately  coincides  with  it.  The  apparent  depth  of  water,  which 

is  only  -r  of  the  real  depth  when  seen  vertically,  is  accordingly  less 

than  -j-  when  seen  obliquely,  and  becomes  a  vanishing  quantity  as 

the  direction  of  vision  approaches  to  parallelism  with  the  surface. 
The  focus  I  determined  in  the  preceding  section  is  at  the  cusp  of  the 
caustic. 

991.  Parallel  Plate. — Rays  falling  normally  on  a  uniform  trans- 
parent plate  with  parallel  faces,  keep  their  course  unchanged;  but 
this  is  not  the  case  with  rays  incident  obliquely.  A  ray  S I  (Fig. 
699),  incident  at  the  angle  S  I  N,  is  refracted  in  the  direction  I  R. 
The  angle  of  incidence  at  R  is  equal  to  the  angle  of  refraction  at  I, 
and  hence  the  angle  of  emergence  S'  R  N'  is  equal  to  the  original 
angle  of  incidence  SIN.  The  emergent  ray  R  S'  is  therefore  parallel 
to  the  incident  ray  S  I,  but  is  not  in  the  same  straight  line  with  it. 


1002 


REFRACTION. 


Objects  seen  obliquely  through  a  plate  are   therefore  displaced 
from  their  true  positions.     Let  S  (Fig.  700)  be  a  luminous  point 


Fig.  699.  -Parallel  Plate. 


Fig.  700.— Vision  through  Plate 


which  sends  light  to  an  eye  not  directly  opposite  to  it,  on  the  other 
side  of  a  parallel  plate.  The  emergent  rays  which  enter  the  eye  are 
parallel  to  the  incident  rays;  but  as  they  have  undergone  lateral 
displacement,  their  point  of  concourse1  is  changed  from  S  to  S',  which 
is  accordingly  the  image  of  S. 

The  displacement  thus  produced  increases  with  the  thickness  of 
the  plate,  its  index  of  refrac- 
tion, and  the  obliquity  of  in- 
cidence. It  furnishes  one  of 
the  simplest  means  of  mea- 
suring the  index  of  refraction 
of  a  substance,  and  is  thus 
employed  in  Pichot's  refrac- 
tometer. 

992.  Multiple  Images  pro- 
duced by  a  Plate.  — Let  S 
(Fig.  701)  be  a  luminous 
point  in  front  of  a  transpar- 
ent plate  with  parallel  faces. 
Of  the  rays  which  it  sends 
to  the  plate,  some  will  be 
reflected  from  the  front,  thus 
giving  rise  to  an  image  S'.  Another  portion  will  enter  the  plate, 

1  The  rays  which  compose  the  pencil  that  enters  the  eye  will  not  exactly  meet  (when 
produced  backwards)  in  any  one  point.  There  will  be  two  focal  lines,  just  as  in  the  case 
of  spherical  mirrors  (§  974,  975). 


Fig.  701.— Multiple  Images  in  Plate. 


ASTRONOMICAL   REFRACTION7. 


1003 


undergo  reflection  at  the  back,  and  emerge  with  refraction  at  the 
front,  giving  rise  to  a  second  image  S°.  Another  portion  will  undergo 
internal  reflection  at  the  front,  then  again  at  the  back,  and  by  emerg- 
ing in  front  will  form  a  third  image  S^  The  same  process  may  be 
repeated  several  times;  and  if  the  luminous  object  be  a  candle,  or  a 
piece  of  bright  metal,  a  number  of  images,  one  behind  another,  will 

be  visible  to  an  eye  properly  placed  in 
front  (Fig.  702).  All  the  successive 
images,  after  the  first  two,  continually 
diminish  in  brightness.  If  the  glass 
be  silvered  at  the  back,  the  second 
image  is  much  brighter  than  the  first, 
when  the  incidence  is  nearly  normal, 
but  as  the  angle  of  incidence  increases, 
the  first  image  gains  upon  the  second, 
and  ultimately  surpasses  it.  This  is 
due  to  the  fact  that  the  reflecting 
power  of  a  surface  of  glass  increases 
with  the  angle  of  incidence. 

If  the  luminous  body  is  at  a  dis- 
tance which  may  be  regarded  as  in- 
finite,— if  it  is  a  star,  for  example, — 
all  the  images  should  coincide,  and 

form  only  a  single  image,  occupying  a  position  which  does  not 
vary  with  the  position  of  the  observer,  provided  that  the  plate 
is  perfectly  homogeneous,  and  its  faces  perfectly  plane  and  par- 
allel. A  severe  test  is  thus  furnished  of  the  fulfilment  of  these 
conditions. 

Plates  are  sometimes  tested,  for  parallelism  and  uniformity,  by 
supporting  them  in  a  horizontal  position  on  three  points,  viewing 
the  image  of  a  star  in  them  with  a  telescope  furnished  with  cross 
wires,  and  observing  whether  the  image  is  displaced  on  the  wires 
when  the  plate  is  shifted  into  a  different  position,  still  resting  on 
the  same  three  points. 

993.  Superimposed  Plates.  Astronomical  Refraction. — We  have 
stated  in  §  986  that  the  relative  index  from  one  medium  into  an- 
other is  equal  to  the  absolute  index  of  the  second  divided  by  that  of 
the  first.  Hence  if  ^  /*2  are  the  absolute  indices,  and  <pt  <p2  the  angles 
which  the  two  parts  of  the  refracted  ray  make  with  the  normal,  we 
have 


Fig.  702. 
Images  of  Candle  in  Looking-glass. 


1004  REFKACTION. 

sin  tpi  =  —  sin  <&, 
Mi 

or 

(1) 


When  a  number  of  plates  are  superimposed,  they  will  have  a  com- 
mon normal.  Let  a  ray  pass  through  them  all;  let  ^  denote  the 
absolute  index  of  any  one  of  the  plates,  and  <p  the  angle  which  the 
portion  of  the  ray  that  lies  in  this  plate  makes  with  the  normal; 
then  equation  (1)  shows  that  ^  sin  <p  will  have  the  same  value  for  all 
parts  of  the  ray.  Hence  if  the  value  of  <P  for  the  first  plate  be  given, 
its  value  for  any  plate  in  the  series  depends  only  on  the  value  of  ^ 
for  that  plate,  and  will  not  be  altered  by  removing  some  or  all  of 
the  intervening  plates. 

This  reasoning  can  be  applied  to  the  transmission  of  a  ray  from  a 
star  through  the  earth's  atmosphere,  if  the  distance  of  the  star  from 
the  zenith  does  not  exceed  20°  or  30°.  The  portion  of  atmosphere 
traversed  may  be  regarded  as  a  series  of  horizontal  plates,  and  the 
slope  of  the  ray  in  the  lowest  plate  will  be  the  same  as  if  all  the 
plates  above  it  were  removed.  In  the  case  of  stars  near  the  horizon, 
the  length  of  the  path  in  air  is  so  great  that  the  curvature  of  the 
earth  cannot  be  left  out  of  account,  in  other  words,  the  layers  tra- 
versed cannot  be  regarded  as  parallel  plates. 

994.  Refraction  through  a  Prism.  —  For  optical  purposes,  any  por- 
tion of  a  transparent  body  lying  between  two  plane  faces  which  are 
not  parallel  may  be  regarded  as  a  prism.1  The  line  in  which  these 
faces  meet,  or  would  meet  if  produced,  is  called  the  edge  of  the  prism, 
and  a  section  made  by  a  plane  perpendicular  to  them  both  is  called 
a  principal  section.  The  prisms  chiefly  employed  are  really  prisms 
in  the  geometrical  sense  of  the  word.  Their  principal  sections  are 
usually  triangular,  and  are  very  frequently  equilateral,  as  in  Fig.  703. 
The  stand  usually  employed  for  prisms  when  mounted  separately  is 
represented  in  Fig.  704.  It  contains  several  joints.  The  uppermost 
is  for  rotating  the  prism  about  its  own  axis.  The  second  is  for  turn- 
ing the  prism  so  that  its  edges  shall  make  any  required  angle  with 
the  vertical.  The  third  gives  motion  about  a  vertical  axis,  and  also 
furnishes  the  means  of  raising  and  lowering  the  prism  through  a  range 
of  several  inches. 

Let  S  I  (Fig.  705)  be  an  incident  ray  in  the  plane  of  a  principal 
section  of  the  prism.  If  the  external  medium  be  air,  or  any  other 

1  This  amounts  to  saying  that  the  word  prism  in  optics  means  wedge. 


PRISLL 


1005 


substance  of  less  refractive  power  than  the  prism,  the  ray  in  entering 
the  prism  will  be  bent  nearer  to  the  normal,  taking  such  a  course  as 
I E,  and  in  leaving  the  prism  will  be  bent  away  from  the  normal, 


Fig.  703.—  Equilateral  Prism. 


Fig.  704.— Prism  mounted  on  Stand. 


taking  the  course  E  B.     The  effect  of  these  two  refractions  is,  there- 
fore, to  turn  the  ray  away  from  the  edge  (or  refracting  angle)  of  the 

prism.  In  practice,  the  prism  is 
usually  so  placed  that  IE,  the  path 
of  the  ray  through  the  prism, 
makes  equal  angles  with  the  two 
faces  at  which  refraction  occurs 
(§  995).  If  the  prism  is  turned 
very  far  from  this  position,  the 
course  of  the  ray  may  be  alto- 
gether different  from  that  repre- 
sented in  the  figure;  it  may,  for 
example,  enter  at  one  face,  be  in- 
ternally reflected  at  another,  and  come  out  at  the  third;  but  we  at 
present  exclude  such  cases  from  consideration. 

The  direction  of  deviation  is  easily  shown  experimentally,  by 
admitting  a  narrow  beam  of  sunlight  into  a  dark  room,  and  intro- 
ducing a  prism  in  its  course.  It  will  be  found  that  the  refracted 


Fig.  705.— Refraction  through  Prism. 


1000  REFRACTION. 

beam,  in  the  circumstances  represented  in  Fig.  705,  is  turned  aside 
some  40°  or  50°  from  its  original  course.1 

Since  the  rays  which  traverse  a  prism  are  bent  away  from  the 
edge,  the  object  from  which  they  proceed  will  appear,  to  an  observer 
looking  through  the  prism,  to  be  more  nearly  in  the  direction  of  the 


Fig.  706.— Vision  through  Prism. 

edge  than  it  really  is.  If,  for  example,  he  looks  at, the  flame  of  a 
candle  through  a  prism  placed  so  that  the  edge  which  corresponds 
to  the  refracting  angle  is  at  the  top  (Fig.  706),  the  apparent  place  of 
the  flame  will  be  above  its  true  place. 

995.  Formula  for  Refraction  through  Prisms.  Minimum  Deviation. 
— Let  S I  (Fig.  707)  be  an  incident  ray  in  the  plane  of  a  principal 
section  A  B  C  of  a  prism.  Let  i  be  the  angle  of  incidence  SIN, and 

1  The  phenomena  here  described  are  complicated  in  practice  by  the  unequal  refrangi- 
bility  of  rays  of  different  colours  (Chap.  Ixxii.).  The  complication  may  be  avoided  by 
employing  homogeneous  light,  of  which  a  spirit-lamp,  with  common  salt  sprinkled  on  the 
wick,  affords  a  nearly  perfect  example. 


REFRACTION   THROUGH   PRISM.  1007 

r  the  angle  of  refraction  H 1 1'.  Then,  denoting  the  index  of  refrac- 
tion by  /u,  we  have  sin  i=p  sin  r.  In  like  manner,  putting  r'  for 
the  angle  of  internal  incidence  on  the  second  face  1 1'  M,  and  i'  for 
the  angle  of  external  refraction  N'  I'  R,  we  have  sin  i'=f*  sin  r. 


Fig.  707.— Refraction  through  Prism. 

The  deviation  produced  at  I  is  i  -  r,  and  that  at  I'  is  i'  -  r',  so  that 
the  total  deviation,  which  is  the  acute  angle  D  contained  between 
the  rays  S  I,  R  I',  when  produced  to  meet  at  o,  is 

T>  =  i-r  +  i'-r'.  (1) 

But  if  we  drop  a  perpendicular  from  the  angular  point  A  on  the  ray 
1 1',  it  will  divide  the  refracting  angle  BAG  into  two  parts,  of  which 
that  on  the  left  will  be  equal  to  r,  and  that  on  the  right  to  r',  since 
the  angle  contained  between  two  lines  is  equal  to  that  contained 
between  their  perpendiculars.  We  have  therefore  A=r+r',  and  by 
substitution  in  the  above  equation 

D=i+i'-A.  (2) 

When  the  path  of  the  ray  through  the  prism  1 1'  makes  equal  angles 
with  the  two  faces,  the  whole  course  of  the  ray  is  symmetrical  with 
respect  to  a  plane  bisecting  the  refracting  angle,  so  that  we  have 

Equation  (2)  thus  becomes 

D  =  2  i- A,  whence  t=—i-,  (3) 

8inA±D 
?ii= ?-.  W 


"»'         sin 

This  last  result  is  of  great  practical  importance,  as  it  enables  us  to 


1008 


EEFRACTION. 


calculate  the  index  of  refraction  p  from  measurements  of  the  refract- 
ing angle  A  of  the  prism,  and  of  the  deviation  D  which  occurs  when 
the  ray  passes  symmetrically. 

When  a  beam  of  sunlight  in  a  dark  room  is  transmitted  through 
a  prism,  it  will  be  found,  on  rotating  the  prism  about  its  axis,  that 
there  is  a  certain  mean  position  which  gives  smaller  deviation  of  the 
transmitted  light  than  positions  on  either  side  of  it;  and  that,  when 
the  prism  is  in  this  position,  a  small  rotation  of  it  has  no  sensible 
effect  on  the  amount  of  deviation.     The  position  determined  experi- 
mentally by  these  conditions,  and  known  as  the  position  of  minimum 
deviation,  is  the  position  in  which  the  ray  passes  symmetrically. 
996.  Construction  for  Deviation. — The  following  geometrical  con- 
struction furnishes  a  very  sim- 
ple method  of  representing  the 
variation  of  deviation  with  the 
angle  of  incidence: — 

1.  When  the  refraction  is  at 
a  single  surface,  describe  two 
circular  arcs  about  a  common 
centre  O  (Fig.  708),  the  ratio 
of  their  radii  being  the  index 
General  Construction  for  Deviation.  of  refraction.  Then  if  the  in- 

cidence  is  from  rare  to  dense, 

draw  a  radius  O  A  of  the  smaller  circle  to  represent  the  direction  of 
the  incident  ray,  and  let  NAB  be  the  direction  of  the  normal  to 
the  surface  at  the  point  of  incidence,  so  that  O  A  N  is  the  angle  of 
incidence.  Join  O  B.  Then  O  B  N  is  the  angle  of  refraction,  since 

=  index  of  refraction;  hence  OB  is  parallel  to  the  re- 
fracted ray.  If  the  incidence  is  from  dense  to  rare,  we  must  draw 
O  B  to  represent  the  incident  ray,  make  0  B  N  equal  to  the  angle 
of  incidence,  and  join  O  A.  In  either  case  the  angle  A  O  B  is  the 
deviation,  and  it  evidently  increases  with  the  angle  of  incidence 
O  A  N,  attaining  its  greatest  value  when  this  angle  (0  A  N"  in  the 
figure)  is  a  right  angle,  in  which  case  the  angle  of  refraction  O  B"  N" 
is  the  critical  angle. 

2.  To  find  the  deviation  in  refraction  through  a  prism,  describe 
two  concentric  circular  arcs  as  before  (Fig.  709),  the  ratio  of  their 
radii  being  the  index  of  refraction.  Draw  the  radius  0  A  of  the 
smaller  circle  to  represent  the  incident  ray,  N  B  to  represent  the 


MINIMUM  DEVIATION   FOR   PRISM. 


1009 


normal  at  the  first  surface,  BN'the  normal  at  the  second  surface.  Then 
O  B  represents  the  direction  of  the  ray  in  the  prism,  O  A'  the  direction 
of  the  emergent  ray,  and  A  O  A'  is  accordingly  the  total  deviation. 
In  fact  we  have 

OAN  —  angle  of  incidence  at  first  surface. 

OBN  =       „        refraction          „ 

OBN'  =        „          incidence  at  second  surface. 

OA'N'  =        „          refraction  „ 

A  O  B  =  deviation  at  first  surface. 

B  O  A'  =        „  second   „ 

ABA'  =.  angle  between  normals '=. angle  of  prism. 

Again,  the  deviation  AOA',  being  the  angle  at  the  centre  of 
a  circle,  is  measured  by  the  arc  A  A',  which  subtends  it.  To  obtain 
the  minimum  deviation,  we  must  so  arrange  matters  that  the  angle 
ABA'  being  given  (= angle  of  prism),  the  arc  A  A'  shall  be  a  mini- 
mum. Let  ABA',  a  B  a'  (Fig.  710),  be  two  consecutive  positions, 


Fig.  703.— Application  to  Prism. 


Fig.  710.— Proof  of  Minimum  Deviation. 


B  A'  and  B  a'  being  greater  than  B  A  and  B  a.  Then,  since  the 
small  angles  A  B  a,  A'  B  a'  are  equal,  it  is  obvious,  for  a  double 
reason,  that  the  small  arc  A'  a  is  greater  than  A  a,  and  hence  the 
whole  arc  a  a'  is  greater  than  A  A'.  The  deviation  is  therefore  in- 
creased by  altering  the  position  in  such  a  way  as  to  make  B  A  and 
B  A'  depart  further  from  equality,  and  is  a  minimum  when  they 
are  equal. 

997.  Conjugate  Foci  for  Minimum  Deviation. — When  the  angle  of 
incidence  is  nearly  that  corresponding  to  minimum  deviation,  a 
small  change  in  this  angle  has  no  sensible  effect  on  the  amount  of 
deviation. 

Hence  a  small  pencil  of  rays  sent  in  this  direction  from  a  luminous 
point,  and  incident  near  the  refracting  edge,  will  emerge  with  their 
divergence  sensibly  unaltered,  so  that  if  produced  backwards  they 
64 


1010 


REFRACTION. 


would  meet  in  a  virtual  focus  at  the  same  distance  (but  of  course 
not  in  the  same  direction)  as  the  point  from  which  they  came. 

In  like  manner,  if  a  small  pencil  of  rays  converging  towards  a 
point,  are  turned  aside  by  interposing  the  edge  of  a  prism  in  the 
position  of  minimum  deviation,  they  will  on  emergence  converge  to 
another  point  at  the  same  distance.  We  may  therefore  assert  that, 
neglecting  the  thickness  of  a  prism,  conjugate  foci  are  at  the  same 
distance  from  it,  and  on  the  same  side,  when  the  deviation  is  a 
minimum. 

998.  Double  Refraction. — Thus  far  we  have  been  treating  of  what 
is  called  single  refraction.  We  have  assumed  that  to  each  given 
incident  ray  there  corresponds  only  one  refracted  ray.  This  is  true 
when  the  refraction  is  into  a  liquid,  or  into  well-annealed  glass,  or 
into  a  crystal  belonging  to  the  cubic  system. 

On  the  other  hand,  when  an  incident  ray  is  refracted  into  a 
crystal  of  any  other  than  the  cubic  system,  or  into  glass  which  is 
unequally  stretched  or  compressed  in  different  directions;  for  example, 
into  unannealed  glass,  it  gives  rise  in  general  to  two  refracted  rays 
which  take  different  paths;  and  this  phenomenon  is  called  double 

refraction.  Attention  was 
first  called  to  it  in  1670  by 
Bartholin,  who  observed 
it  in  the  case  of  Iceland- 
spar,  and  its  laws  for  this 
substance  were  accurately 
determined  by  Huygens. 

999.  Phenomena  of  Double 
Refraction  in  Iceland-spar. 
—  Iceland-spar  or  calc- 
spar  is  a  form  of  crystal- 
lized carbonate  of  lime, 
and  is  found  in  large  quan- 
tity in  the  country  from 
which  it  derives  its  name.  It  is  usually  found  in  rhombohedral  form, 
as  represented  in  Figs.  711,  712. 

To  observe  the  phenomenon  of  double  refraction,  a  piece  of  the 
spar  may  be  laid  on  a  page  of  a  printed  book.  All  the  letters  seen 
through  it  will  appear  double,  as  in  Fig.  712;  and  the  depth  of  their 
blackness  is  considerably  less  than  that  of  the  originals,  except  where 
the  two  images  overlap. 


Fig.  711. — Iceland-spar. 


DOUBLE   REFRACTION.  1011 

In  order  to  state  the  laws  of  the  phenomena  with  precision,  it  is 
necessary  to  attend  to  the  crystalline  form  of  Iceland-spar. 

At  the  corner  which  is  represented  as  next  us  in  Fig.  711  three 
equal  obtuse  angles  meet;  and  this  is  also  the  case  at  the  opposite 


Fig.  712.— Double  Refraction  of  Iceland-spar. 

corner  which  is  out  of  sight.  If  a  line  be  drawn  through  one  of 
these  corners,  making  equal  angles  with  the  three  edges  which  meet 
there,  it  or  any  line  parallel  to  it  is  called  the  axis  of  the  crystal; 
the  axis  being  properly  speaking  not  a  definite  line  but  a  definite 
direction. 

The  angles  of  the  crystal  are  the  same  in  all  specimens;  but  the 


Fig.  713.— Axis  of  the  Crystal 

lengths  of  the  three  edges  (which  may  be  called  the  oblique  length, 
breadth,  and  thickness)  may  have  any  ratios  whatever.  If  the  crystal 
is  of  such  proportions  that  these  three  edges  are  equal,  as  in  the  first 
part  of  Fig.  713,  the  axis  is  the  direction  of  one  of  its  diagonals, 
which  is  represented  in  the  figure. 

Any  plane  containing  (or  parallel  to)  the  axis  is  called  a  principal 
plane  of  the  crystal. 

If  the  crystal  is  laid  over  a  dot  on  a  sheet  of  paper,  and  is  made 


1012  REFRACTION. 

to  rotate  while  remaining  always  in  contact  with  the  paper,  it  will 
be  observed  that,  of  the  two  images  of  the  dot,  one  remains  un- 
moved, and  the  other  revolves  round  it.  The  former  is  called  the 
ordinary,  and  the  latter  the  extraordinary  image.  It  will  also  be 
observed. that  the- former  appears  nearer  than  the  latter,  being  more 
lifted  up  by  refraction. 

The  rays  which  form  the  ordinary  image  follow  the  ordinary  law 
of  sines  (§  983).  They  are  called  the  ordinary  rays.  Those  which 
form  the  extraordinary  image  (called  the  extraordinary  rays)  do  not 
follow  the  law  of  sines,  except  when  the  refracting  surface  is  parallel 
to  the  axis,  and  the  plane  of  incidence  perpendicular  to  the  axis;  and 
in  this  case  their  index  of  refraction  (called  the  extraordinary  index) 
is  different  from  that  of  the  ordinary  rays.  The  ordinary  index  is 
1*65,  and  the  extraordinary  T48. 

When  the  plane  of  incidence  is  parallel  to  the  axis,  the  extra- 
ordinary ray  always  lies  in  this  plane,  whatever  be  the  direction  of 
the  refracting  surface;  but  the  ratio  of  the  sines  of  the  angles  of 
incidence  and  refraction  is  variable. 

When  the  plane  of  incidence  is  oblique  to  the  axis,  the  extra- 
ordinary ray  generally  lies  in  a  different  plane. 

We  shall  recur  to  the  subject  of  double  refraction  in  the  concluding 
chapter  of  this  volume. 


CHAPTER    LXX. 


LENSES. 


1000.  Forms  of  Lenses. — A  lens  is  usually  a  piece  of  glass  bounded 
by  two  surfaces  which  are  portions  of  spheres.  There  are  two  prin- 
cipal classes  of  lenses. 

1.  Converging  lenses  or  convex  lenses,  which  have  one  or  other  of 
the  three  forms  represented  in  Fig.  714.  The  first  of  these  is  called 
double  convex,  the  second  plano-convex,  and  the  third  concavo- 
convex.  This  last  is  also  called  a  converging  meniscus.  All  three 


Fig.  714.— Converging  Lenses.  Fig.  715.— Diverging  Lenses. 

are  thicker  in  the  middle  than  at  the  edges.  They  are  called  con- 
verging, because  rays  are  always  more  convergent  or  less  divergent 
after  passing  through  them  than  before. 

2.  Diverging  lenses  or  concave  lenses  (Fig.  715)  produce  the  oppo- 
site effect,  and  are  characterized  by  being  thinner  in  the  middle  than 
at  the  edges.  Of  the  three  forms  represented,  the  first  is  double  con- 
cave, the  second  plano-concave,  and  the  third  convexo-concave  (also 
called  a  diverging  meniscus). 


1014 


LENSES. 


Fig.  716.— Principal  Focus  of  Convex  Lens. 


From  the  immense  importance  of  lenses,  especially  convex  lenses, 
'in  practical  optics,  it  will  be  necessary  to  explain  their  properties  at 
some  length. 

1001.  Principal  Focus. — A  lens  is  usually  a  solid  of  revolution,  and 
the  axis  of  revolution  is  called  the  axis  of  the  lens,  or  sometimes  the 
principal  axis.  When  the  surfaces  are  spherical,  it  is  the  line  join- 
ing their  centres  of  curvature. 

When  rays  which  were  originally  parallel  to  the  principal  axis 

pass  through  a  convex 
lens  (Fig.  716),  the  ef- 
fect of  the  two  refrac- 
tions which  they  un- 
dergo, one  on  entering 
and  the  other  on  leav- 
ing the  lens,  is  to  make 
them  all  converge  ap- 
proximately to  one 
point  F,  which  is 
called  the  principal  focus.  The  distance  A  F  of  the  principal  focus 
from  the  lens  is  called  the  principal  focal  distance,  or  more  briefly 
and  usually,  the  focal  length  of  the  lens.  There  is  another  prin- 
cipal focus  at  the  same  distance  on  the  other  side  of  the  lens,  cor- 
responding to  an  inci- 
dent beam  coming  in  the 
opposite  direction.  The 
focal  length  depends  on 
the  convexity  of  the  sur- 
faces of  the  lens,  and  also 
on  the  refractive  power 
of  the  material  of  which 
it  is  composed,  being 
shortened  either  by  an 

increase  of  refractive  power  or  by  a  diminution  of  the  radii  of  cur- 
vature of  the  faces. 

In  the  case  of  a  concave  lens,  rays  incident  parallel  to  the  principal 
axis  diverge  after  passing  through;  and  their  directions,  if  produced 
backwards,  would  approximately  meet  in  a  point  F  (Fig.  717),  which 
is  still  called  the  principal  focus.  It  is  only  a  virtual  focus,  inasmuch 
as  the  emergent  rays  do  not  actually  pass  through  it,  whereas  the 
principal  focus  of  a  converging  lens  is  real. 


Fig.  717.— Principal  Focus  of  Concave  Lens. 


CENTRE   OF  LENS. 


1015 


Kg.  718.— Centre  oi  Lena. 


1002.  Optical  Centre  of  a  Lens.  Secondary  Axes.— Let  0  and  0' 
(Fig.  718)  be  the  centres  of  the  two  spherical  surfaces  of  a  lens. 
Draw  any  two  parallel  radii  O  I, 
O'  E  to  meet  these  surfaces,  and 
let  the  joining  line  I  E  represent 
a  ray  passing  through  the  lens. 
This  ray  makes  equal  angles  with 
the  normals  at  I  and  E,  sirice 
these  latter  are  parallel  by  con- 
struction ;  hence  the  incident  and 
emergent  rays  S  I,  E  R  also  make 
equal  angles  with  the  normals, 
and  are  therefore  parallel.  In 
fact,  if  tangent  planes  (indicated 
by  the  dotted  lines  in  the  figure) 
are  drawn  at  I  and  E,  the  whole 

course  of  the  ray  SIER  will  be  the  same  as  if  it  had  passed  through 
a  plate  bounded  by  these  planes. 

Let  C  be  the  point  in  which  the  line  I E  cuts  the  principal  axis, 
and  let  R,  R'  denote  the  radii  of  the  two  spherical  surfaces.  Then, 
from  the  similarity  of  the  triangles  0  C  I,  0'  C  E,  we  have 

00       R  m 

CO'  ~  R" 

which  shows  that  the  point  C  divides  the  line  of  centres  0  0'  in  a 
definite  ratio  depending  only  on  the  radii.  Every  ray  whose  direc- 
tion on  emergence  is  parallel  to  its  direction  before  entering  the  lens, 
must  pass  through  the  point  C  in  traversing  the  lens;  and  conversely, 
every  ray  which,  in  its  course  through  the  lens,  traverses  the  point 
C,  has  parallel  directions  at  incidence  and  emergence.  The  point  C 
which  possesses  this  remarkable  property  is  called  the  centre,  or 
optical  centre,  of  the  lens. 

In  the  case  of  a  double  convex  or  double  concave  lens,  the  optical 
centre  lies  in  the  interior,  its  distances  from  the  two  surfaces  being 
directly  as  their  radii.  In  plano-convex  and  plano-concave  lenses  it 
is  situated  on  the  convex  or  concave  surface.  In  a  meniscus  of  either 
kind  it  lies  outside  the  lens  altogether,  its  distances  from  the  surfaces 
being  still  in  the  direct  ratio  of  their  radii  of  curvature.1 


These  consequences  follow  at  once  from  equation  (1);  for  the  distances  of  C  from  the 


1016 


LENSES. 


In  elementary  optics  it  is  usual  to  neglect  the  thickness  of  the  lens. 
The  incident  and  emergent  rays  S  I,  E  R  may  then  be  regarded  as 
lying  in  one  straight  line  which  passes  through  C,  and  we  may  lay 
down  the  proposition  that  rays  which  pass  through  the  centre  of  a 
lens  undergo  no  deviation.  Any  straight  line  through  the  centre 
of  a  lens  is  called  a  secondary  axis. 

The  approximate  convergence  of  the  refracted  rays  to  a  point, 
when  the  incident  rays  are  parallel,  is  true  for  all  directions  of  in- 


Fig.  719. — Principal  Focus  on  Secondary  Axis. 

cidence;  and  the  point  to  which  the  emergent  rays  approximately 
converge  (/,  Fig.  719)  is  always  situated  on  the  secondary  axis  (acf) 
parallel  to  the  incident  rays.  The  focal  distance  is  sensibly  the 
same  as  for  rays  parallel  to  the  principal  axis,  unless  the  obliquity 
is  considerable. 
1003.  Conjugate  Foci. — When  a  luminous  point  S  sends  rays  to  a 


Fig.  720.— Conjugate  Foci,  both  Real. 

lens  (Fig.  720),  the  emergent  rays  converge  (approximately)  to  one 

two  faces  are  respectively  the  difference  between  R  and  0  C,  and  the  difference  between 
II'  and  0'  C,  and  we  have 

R    _  OC  _  B-OC 

If  ~  0*0  ~  R— 6'C' 


FORMULA   FOR   LENSES.  1017 

point  S';  whence  it  follows  that  rays  sent  from  S'  to  the  lens  would 
converge  (approximately)  to  S.  Two  points  thus  related  are  called 
conjugate  joci  of  the  lens,  and  the  line  joining  them  always  passes 
through  the  centre  of  the  lens;  in  other  words,  they  must  either  be 
both  on  the  principal  axis,  or  both  on  the  same  secondary  axis. 

The  fact  that  rays  which  come  from  one  point  go  to  one  point  is 
the  foundation  of  the  theory  of  images,  as  we  have  already  explained 
in  connection  with  mirrors  (§  967). 

The  diameters  of  object  and  image  are  directly  as  their  distances 
from  the  centre  of  the  lens,  and  the  image  will  be  erect  or  inverted 
according  as  the  object  and  image  lie  on  the  same  side  or  on  opposite 
sides  of  this  centre  (§  971).  There  is  also,  in  the  case  of  lenses,  the 
same  difference  between  an  image  seen  in  mid-air  and  an  image 
thrown  on  a  screen  which  we  have  pointed  out  in  §  974. 

It  is  to  be  remarked  that  the  distinction  between  principal  and 
secondary  axes  has  much  more  significance  in  the  case  of  lenses  than 
of  mirrors;  and  images  produced  by  a  lens  are  more  distinct  in  the 
neighbourhood  of  the  principal  axis  than  at  a  distance  from  it. 

1004.  Formulae  relating  to  Lenses. — The  deviation  produced  in  a 
ray  by  transmission  through  a  lens  will  not  be  altered  by  substituting 


Fig.  721.— Diagram  showing  Path  of  Ray,  and  Normals. 

for  the  lens  a  prism  bounded  by  planes  which  touch  the  lens  at  the 
points  of  incidence  and  emergence;  and  in  the  actual  use  of  lenses, 
the  direction  of  the  rays  with  respect  to  the  supposed  prism  is  such 
as  to  give  a  deviation  not  differing  much  from  the  minimum.  The 
expression  for  the  minimum  deviation  (§  995)  is2i-2ror2i~A; 
and  when  the  angle  of  the  prism  is  small,  as  it  is  in  the  case  of 

ordinary  lenses,  we  may  assume  -sr-jj— ss  p;  so  that  2  i  becomes 
2  p.  r  or  ft  A,  and  the  expression  for  the  deviation  becomes 
(M-l)A,  (1) 


1018  LENSES. 

A  being  the  angle  between  the  tangent  planes  (or  between  the 
normals)  at  the  points  of  entrance  and  emergence. 

Let  xl  and  xz  denote  the  distances  of  these  points  respectively  from 
the  principal  axis,  and  rv  r2  the  radii  of  curvature  of  the  faces  on 

which  they  lie.     Then  ^->  —  are  the  sines  of  the  angles  which  the 

normals  make  with  the  axis,  and  the  angle  A  is  the  sum  or  differ- 
ence of  these  two  angles,  according  to  the  shape  of  the  lens.  In  the 
case  of  a  double  convex  lens  it  is  their  sum,  and  if  we  identify  the 
sines  of  these  small  angles  with  the  angles  themselves,  we  have 

A=3+?!.  (2) 

But  if  plt  p.,  denote  the  distances  from  the  faces  of  the  lens  to  the 
points  where  the  incident  and  emergent  rays  cut  the  principal  axis, 

— '  —  are  the  sines  of  the  angles  which  these  rays  make  with  the 
Pi  Pi 

axis,  and  the  deviation  is  the  sum  or  difference  of  these  two  angles, 
according  as  the  conjugate  foci  are  on  opposite  sides  or  on  the  same 
side  of  the  lens.  In  the  former  case,  identifying  the  angles  with 

their  sines,  the  deviation  is  — +  — ,  and  this,  by  formula  (1),  is  to  be 
equal  to  (p-1)  A,  that  is,  to  (/*-!)  (^+^). 

If  the  thickness  of  the  lens  is  negligible  in  comparison  with  pt,  p.2> 
we  may  regard  xl  and  x.2  as  equal,  and  the  equation 


will  reduce  to 

If  pl  is  infinite,  the  incident  rays  are  parallel,  and  p2  is  the  principal 
focal  length,  which  we  shall  denote  by  /.     We  have  therefore 

Y=(ft-l)  (^-  +  ^)  (5) 

and 

111  IR\ 

—  +  —  -—.  (6) 

Pi    P*    f 

1005.  Conjugate  Foci  on  Secondary  Axis. — Let  M  (Fig.  722)  be  a 
luminous  point  on  the  secondary  axis  M  0  M',  O  being  the  centre  of 


FOCI  FOR   OBLIQUE   PENCILS.  1019 

the  lens,  and  let  M'  be  the  point  in  which  an  emergent  ray  corre- 
sponding to  the  incident  ray  M  I  cuts  this  axis.  Let  x  denote  xl  or 
x2,  the  distances  of  the  points  of  incidence  and  emergence  from  the 


Fig.  722. — Conjugate  Foci  on  Secondary  Axis. 

principal  axis,  and  0  the  obliquity  of  the  secondary  axis;  then  x  cos  0 
is  the  length  of  the  perpendicular  from  I  upon  M  M',  and  x^\  '  -^>-j  ' 

are  the  sines  of  the  angles  OMI,  OM'I  respectively.  But  the 
deviation  is  the  sum  of  these  angles;  hence,  proceeding  as  in  last 
section,  we  have 

\         X  /7> 

)  =  7 

The  fact  that  x  does  not  appear  in  equations  (6)  and  (8)  shows  that, 
for  every  position  of  a  luminous  point,  there  is  a  conjugate  focus, 
lying  on  the  same  axis  as  the  luminous  point  itself.  Equation  (8) 
shows  that  the  effective  focal  length  becomes  shorter  as  the  obliquity 
becomes  greater,  its  value  being  /  cos  0,  where  0  is  the  obliquity. 

If  we  take  account  of  the  fact  that  the  rays  of  an  oblique  pencil 
make  the  angles  of  incidence  and  emergence  more  unequal  than  the 
rays  of  a  direct  pencil  and  thus  (by  the  laws  of  prisms)  undergo 
larger  deviation,  we  obtain  a  still  further  shortening  of  the  effective 
focal  length  for  oblique  pencils. 

When  the  obliquity  is  small,  cos  0  may  be  regarded  as  unity,  and 
we  may  employ  the  formula 

&*-•) 

for  oblique  as  well  as  for  direct  pencils. 

1006.  Discussion  of  the  Formula  for  Convex  Lenses. — For  convex 


1020  LENSES. 

lenses /is  to  be  regarded  as  positive;  p  will  be  positive  when  measured 
from  the  lens  towards  the  incident  light,  and  p'  when  measured  in 
the  direction  of  the  emergent  light. 

Formula  (6),  being  identical  with  equation  (6)  of  §  968,  leads  to 
results  analogous  to  those  already  deduced  for  concave  mirrors. 

As  one  focus  advances  from  infinite  distance  to  a  principal  focus, 
its  conjugate  moves  away  from  the  other  principal  focus  to  infinite 
distance  on  the  other  side.  The  more  distant  focus  is  always  moving 
more  rapidly  than  the  nearer,  and  the  least  distance  between  them 
is  accordingly  attained  when  they  are  equidistant  from  the  lens;  in 
which  case  the  distance  of  each  of  them  from  the  lens  is  2  /,  and  their 
distance  from  each  other  4/. 

If  either  of  the  distances,  as  p,  is  less  than  /,  the  formula  shows 
that  the  other  distance  p  is  negative.  The  meaning  is  that  the  two 

foci  are  on  the  same 
side  of  the  lens,  and  in 
this  case  one  of  them 
(the  more  distant  of  the 
two)  must  be  virtual. 
For  example,  in  Fig. 
723,  if  S,  S'  are  a  pair 
of  conjugate  foci,  one 
of  them  S  being  be- 

Fig.  723.-ConjuSate  Foci,  one  Real,  one  Virtual  tween       the       principal 

focus  F  and   the   lens, 

rays  sent  to  the  lens  by  a  luminous  point  at  S,  will,  after  emergence, 
diverge  as  if  from  S';  and  rays  coming  from  the  other  side  of  the 
lens,  if  they  converge  to  S'  before  incidence,  will  in  reality  be  made 
to  meet  in  S.  As  S  moves  towards  the  lens,  S'  moves  in  the  same 
direction  more  rapidly;  and  they  become  coincident  at  the  surface 

of  the  lens.  The  formula  in  fact  shows  that  if  -  is  very  great  in 
comparison  with  7,  and  positive,  —f  must  be  very  great  and  negative; 

that  is  to  say,  if  p  is  a  very  small  positive  quantity,  p  is  a  very  small 
negative  quantity. 

1007.  Formation  of  Real  Images.— Let  A  B  (Fig.  724)  be  an  object 
in  front  of  a  lens,  at  a  distance  exceeding  the  principal  focal  length. 
It  will  have  a  real  image  on  the  other  side  of  the  lens.  To  deter- 
mine the  position  of  the  image  by  construction,  draw  through  any 
point  A  of  the  object  a  line  parallel  to  the  principal  axis,  meeting 


SIZE   OF  IMAGE. 


1021 


724.— Real  and  Diminished  Image. 


the  lens  in  A'.  The  ray  represented  by  this  line  will  after  refrac- 
tion, pass  through  the  principal  focus  F;  and  its  intersection  with 
the  secondary  axis  A  O 
determines  the  position 
of  a,  the  focus  conju- 
gate to  A.  We  can  in 
like  manner  determine 
the  position  of  b,  the 
focus  conjugate  to  B, 
another  point  of  the 
object;  and  the  joining 
line  ab  will  then  be 
the  image  of  the  line 

AB.     It  is  evident  that  if  a 6  were  the  object,  AB  would  be  the 
image. 

Figs.  724,  725  represent  the  cases  in  which  the  distance  of  the 
object     is    respectively 
greater   and   less   than 
twice  the  focal  length 
of  the  lens. 

1008.  Size  of  Image. 
— In  each  case  it  is  evi- 
dent that  ^y =-%"•=*' 
or  the  linear  dimen- 
sions Of  object  and  Fig.  725.-Real  and  Magnified  Image. 

image  are  directly  as 
their  distances  from  the  centre  of  the  lens. 
Again,  since  by  equation  (6) 


we  have 
and 


_ 
p     f     P     pf 


p  _P-f 
p'~  f 


- 
P-f 


(9) 


(10) 


from  which  formula  the  size  of  the  image  can  be  calculated  without 
finding  its  position. 

1009.  Example. — A  straight  lino  25mm-  long  is  placed  perpendi- 


1022  LENSES. 

cularly  on  the  axis,  at  a  distance  of  35  centimetres  from  a  lens  of 
15  centimetres'  focal  length;  what  are  the  position  and  magnitude  of 
the  image? 

To  determine  the  distance  p'  we  have 


For  the  length  of  the  image  we  have 


1010.  Image  on  Cross-wires.  —  The  position  of  a  real  image  seen  in 
mid-air  can  be  tested  by  means  of  a  cross  of  threads,  or  other  con- 
venient mark,  so  arranged  that  it  can  be  fixed  at  any  required  point. 
The  observer  must  fix  this  cross  so  that  it  appears  approximately  to 
coincide  with  a  selected  point  of  the  image.      He  must  then  try 
whether  any  relative  displacement  of  the  two  occurs  on  shifting  his 
eye  to  one  side.     If  so,  the  cross  must  be  pushed  nearer  to  the  lens, 
or  drawn  back,  according  to  the  nature  of  the  observed  displacement, 
which  follows  the  general  rule  of  parallactic  displacement,  that  the 
more  distant  object  is  displaced  in  the  same  direction  as  the  ob- 
server's eye.     The  cross  may  thus  be  brought  into  exact  coincidence 
with  the  selected  point  of  the  image,  so  as  to  remain  in  apparent 
coincidence  with  it  from  all  possible  points  of  view.     When  this 
coincidence  has  been  attained,  the  cross  is  at  the  focus  conjugate  to 
that  which  is  occupied  by  the  selected  point  of  the  object. 

By  employing  two  crosses  of  threads,  one  to  serve  as  object,  and 
the  other  to  mark  the  position  of  the  image,  it  is  easy  to  verify  the 
fact  that  when  the  second  cross  coincides  with  the  image  of  the  first, 
the  first  also  coincides  with  the  image  of  the  second. 

1011.  Aberration  of  Lenses.—  In  the  investigations  of  §§  1004,  1005, 
we  made  several  assumptions  which  were  only  approximately  true. 
The  rays  which  proceed  from  a  luminous  point  to  a  lens  are  in  fact 
not  accurately  refracted  to  one  point,  but  touch  a  curved  surface 
called  a  caustic.    The  cusp  of  this  caustic  is  the  conjugate  focus,  and 
is  the  point  at  which  the  greatest  concentration  of  light  occurs.     It 
is  accordingly  the  place  where  a  screen  must  be  set  to  obtain  the 
brightest  and  most  distinct  image.     Rays  from  the  central  parts  of 
the  lens  pass  very  nearly  through  it;  but  rays  from  the  circumferen- 


CONCAVE  LENS. 


1023 


tial  portions  fall  short  of  it.  This  departure  from  exact  concurrence 
is  called  aberration.  The  distinctness  of  an  image  on  a  screen  is 
improved  by  employing  an  annular  diaphragm  to  cut  off  all  except 
the  central  rays;  but  the  brightness  is  of  course  diminished. 

By  holding  a  convex  lens  in  a  position  very  oblique  to  the 
incident  light,  a  primary  and  secondary  focal  line  can  be  exhibited 
on  a  screen  perpendicular  to  the  beam,  just  as  in  the  case  of  concave 
mirrors  (§  975).  The  experiment,  however,  is  rather  more  difficult 
of  performance. 

1012.  Virtual  Images. — Let  an  object  AB  be  placed  between  a 
convex  lens  and  its  principal  focus.  Then  the  foci  conjugate  to  the 
points  A,  B  are  virtual, 
and  their  positions  can 
be  found  by  construc- 
tionfrom  the  considera- 
tion that  rays  through 
A,  B..  parallel  to  the 
principal  axis,  will  be 
refracted  to  F,  the  prin- 
cipal focus  on  the  other 
side.  These  refracted 
rays,  if  produced  back- 
ward, must  meet  the  secondary  axes  O  A,  O  B  in  the  required  points. 
An  eye  placed  on  the  other  side  of  the  lens  will  accordingly  see  a 
virtual  image,  erect,  magnified,  and  at  a  greater  distance  from  the 
lens  than  the  object.  This  is  the  principle  of  the  simple  microscope. 
The  formula  for  the  distances  D,  d  of  object  and  image  from  the 
lens,  when  both  are  on  the  same  side,  is 


Fig.  726.— Virtual  Image  formed  by  Convex  Lens. 


1        1       1 

D— 3=7' 


(11) 


/  denoting  the  principal  focal  length. 

1013.  Concave  Lens. — For  a  concave  lens,  if  the  focal  length  be  still 
regarded  as  positive,  and  denoted  by  /,  and  if  the  distances  D,  d  be 
on  the  same  side  of  the  lens,  the  formula  becomes 

i-i  =  i  <12> 


which  shows  that  d  is  always  less  than  D;  that  is,  the  image  is 
nearer  to  the  lens  than  the  object. 


LENSES. 


In  Fig.  727,  A  B  is  the  object,  and  a  b  the  image.     Rays  incident 
from  A  and  B  parallel  to  the  principal  axis  will  emerge  as  if  they 

came  from  the  princi- 
pal focus  F.  Hence 
the  points  a  b  are  de- 
termined by  the  in- 
tersections of  the  dot- 
ted lines  in  the  figure 
with  the  secondary 
axes  0  A,  OB.  An 
eye  on  the  other  side 
of  the  lens  sees  the 
image  a  b,  which  is  always  virtual,  erect  and  diminished. 

1014.  Focometer. — Silbermann's  focometer  (Fig.  728)  is  an  instru- 
ment for  measuring  the  focal  lengths  of  convex  lenses,  and  is  based 


Elg.  727.— Virtual  Image  formed  by  Concave  Lens. 


Fig.  728.— Silbermann's  Focometer. 

on  the  principle  (§  1006)  that,  when  the  object  and  its  image  are 
equidistant  from  the  lens,  their  distance  from  each  other  is  four 

times  the  focal  length. 
It  consists  of  a  gradu- 
ated rule  carrying  three 
runners  M,  L,  M'.   The 
middle  one  L  is  the  sup- 
port for  the  lens  which 
is  to  be  examined;  the 
other  two,  MM',  con- 
tain two  thin  plates  of 
horn  or  other  translu- 
cent material,  ruled  with  lines,  which  are  at  the  same  distance  apart 
in  both.     The  sliders  must  be  adjusted  until  the  image  of  one  of 
these  plates  is  thrown«upon  the  other  plate,  without  enlargement  or 


Fig.  729. 


REFRACTION  AT  SPHERICAL   SURFACE  1025 

diminution,  as  tested  by  the  coincidence  of  the  ruled  lines  of  the 
image  with  those  of  the  plate  on  which  it  is  cast.  The  distance 
between  M  and  M'  is  then  read  off,  and  divided  by  4. 

1015.  Refraction  at  a  Single  Spherical  Surface. — Suppose  a  small 
pencil  of  rays  to  be  incident  nearly  normally  upon  a  spherical  sur- 
face which  forms  the  boundary  between  two  media  in  which  the 
indices  are  /^  and  /J2  respectively.  Let  C  (Fig.  729)  be  the  centre 
of  curvature,  and  C  A  the  axis.  Let  Pl  be  the  focus  of  the  incident, 
and  P2  of  the  refracted  rays.  Then  for  any  ray  Px  B,  GBP!  is  the 
angle  of  incidence  and  C  B  P2  the  angle  of  refraction.  Hence  by  the 
law  of  sines  we  have  (§  993) 

Hi  sin  C BPi    =    to  sin  C  B  Pa. 

Dividing  by  sin  BOA,  and  observing  that 

sin  C  B  Pi        C  Pj        C  Pj . 


Dm  ±* \j  A.        j3 IP i       A. IT  i 

sin  C  B  P8       C  Pa       C  P2 1]1HTt.atplv . 

=-— x. —  rr  =r-.^-  —  • ultimately  : 

sin  BOA        BPa~APi,  J' 


we  obtain  the  equation 


which  expresses  the  fundamental  relation  between  the  positions  of 
the  conjugate  foci. 

Let  A.C=r,  APj— pv  A.Pz=p&  then  equation  (13)  becomes 


or,  dividing  by  r, 
which  may  be  written 


£-£.,&=&  (iq 

Again,  let  CA=jo,  CP^qi,  C~P2=q2,  then  equation  (13)  gives 

/>-?i  ~~     p-q* 
or 

1  ^1'  =  1  P^,  (16) 

an  equation  closely  analogous  to  (14)  and  leading  to  the  result 
(analogous  to  (15)) 

ft  ft  ~  Mi  2i  ~  \  Mi  ~  Mi  /  P 

The  signs  of  pv  p2)  r,  in  (14)  and  (15)  are  to  be  determined  by  the 
05 


1026  LENSES. 

rule  that,  if  one  of  the  three  points  Pu  P2,  C  lies  on  the  opposite  side 
of  A  from  the  other  two,  its  distance  from  A  is  to  be  reckoned  op- 
posite in  sign  to  theirs. 

In  like  manner  the  signs  of  qlf  qz,  p,  in  (16)  and  (17)  are  to  be 
determined  by  the  rule  that,  if  one  of  the  three  points  PI}  P2,  A  lies 
on  the  opposite  side  of  C  from  the  other  two,  its  distance  from  C 
is  to  be  reckoned  opposite  in  sign  to  theirs. 

It  is  usual  to  reckon  distances  positive  when  measured  towards 
the  incident  light;  but  the  formulae  will  remain  correct  if  the  opposite 
convention  be  adopted. 

If  /  denote  the  principal  focal  length,  measured  from  A,  we  have, 
by  (15),  writing  /  for  p2  and  making^  infinite, 


and  (15)  may  now  be  written 

t*      On.  -  f*» 
p*  ~  PI  ~  f  ' 

it  being  understood  that  the  positive  direction  for  /  is  the  same  as 
for  plf  pv  and  r. 

The  application  of  these  formulas  to  lenses  in  cases  where  the 
thickness  of  the  lens  cannot  be  neglected,  may  be  illustrated  by  the 
following  example. 

1016.  To  find  the  position  of  the  image  formed  by  a  spherical  lens. 

Let  distances  be  measured  from  the  centre  of  the  sphere,  and  be 
reckoned  positive  on  the  side  next  the  incident  light. 

Then,  if  x  denote  the  distance  of  the  object,  y  the  distance  of  the 
image  formed  by  the  first  refraction,  z  the  distance  of  the  image 
formed  by  the  second  refraction,  a  the  radius  of  the  sphere,  and  p  its 
index  of  refraction;  we  have,  at  the  first  surface, 
p  —  a  ^1  =  1  t^i.  —  /*, 

and  at  the  second  surface 

p=  -  a  Pi  =  P  /*»  =  !. 

Hence  equation  (17)  gives,  for  the  first  refraction, 

L.!_(!_0  I, 

Hy       x         \  fj.  I    a 

and  for  the  second  refraction, 

1     1-     (  -,     l\l-(L    iH 

2-/ty--V1~M/a~\M/<»' 

By  adding  these  two  equations,  we  obtain 


CAMERA   OBSCURA. 


1027 


If  the  incident  rays  are  parallel,  we  have  x  infinite  and  z  = 

— TI  \  > tnat  ig  to  sav>  the  principal  focus  is  at  a  distance  ^j  ~  from 

the  centre,  on  the  side  remote  from  the  incident  light. 

1017.  Camera  Obscura. — The  images  obtained  by  means  of  a  hole  in 

the  shutter  of  a  dark  room  (§  938)  become  sharper  as  the  size  of  the 

hole  is  diminished;  but  this  diminution  in- 
volves loss  of  light,  so  that  it  is  impossible  by 

this  method  to  obtain  an  image  at  once  bright 

and  sharp.     This  difficulty  can  be  overcome 

by  employing  a  lens.     If  the  objects  in  the 

external  landscape  depicted  are  all  at  distances 

many  times  greater  than  the  focal  length  of 

the  lens,  their  images  will  all  be  formed  at 

sensibly  the  same  distance  from  the  lens,  and 

may  be  received  upon  a  screen  placed  at  this 

distance. 
The  images 
thus  ob- 
tained are  - 

inverted,    Fig.  730. -Objective  of  Camera. 

and  are  of 

the   same   size   as    if    a   simple 

aperture  were  employed  instead 

of  a  lens.     This  is  the  principle 

on  which  the  camera  obscura  is 

constructed. 

It  is  a  kind  of  tent  surrounded 
by  opaque  curtains,  and  having 
at  its  top  a  revolving  lantern, 
containing  a  lens  with  its  axis 
horizontal,  and  a  mirror  placed 
behind  it  at  a  slope  of  45°,  to  re- 
flect the  transmitted  light  down- 
wards on  to  a  sheet  of  white  paper 
lying  on  the  top  of  a  table.  Im- 
ages of  external  objects  are  thus 
depicted  on  the  paper,  and  their 
outlines  can  be  traced  with  a 

pencil  if  desired.     It  is  still  better  to  combine  lens  and  mirror  in 
one,  by  the  arrangement  represented  in  section  in  Fig.  730.     Rays 


Fig.  731.— Photographic  Camera. 


1028  LENSES. 

from  external  objects  are  first  refracted  at  a  convex  surface,  then 
totally  reflected  at  the  back  of  the  lens,  which  is  plane,  and  finally 
emerge  through  the  bottom  of  the  lens,  which  is  concave,  but  with 
a  larger  radius  of  curvature  than  the  first  surface.  The  two  refrac- 
tions produce  the  effect  of  a  converging  mensicus.  The  instrument 
is  now  only  employed  for  purposes  of  amusement. 

1018.  Photographic  Camera. — The  camera  obscura  employed  by 
photographers  (Fig.  731)  is  a  box  M  N  with  a  tube  A  B  in  front, 
containing  an  object-glass  at  its  extremity.  The  object-glass  is 
usually  compound,  consisting  of  two  single  lenses  E,  L,  an  arrange- 
ment which  is  very  commonly  adopted  in  optical  instruments,  and 
which  has  the  advantage  of  giving  the  same  effective  focal  length  as 
a  single  lens  of  smaller  radius  of  curvature,  while  it  permits  the 
employment  of  a  larger  aperture,  and  consequently  gives  more  light. 
At  G  is  a  slide  of  ground  glass,  on  which  the  image  of  the  scene  to 
be  depicted  is  thrown,  in  setting  the  instrument.  The  focussing  is 
performed  in  the  first  place  by  sliding  the  part  M  of  the  box  in  the 
part  N,  and  finally  by  the  pinion  V  which  moves  the  lens.  When 
the  image  has  thus  been  rendered  as  sharp  as  possible,  the  sensitized 
plate  is  substituted  for  the  ground  glass.1 


1  The  photographic  processes  at  present  in  use  are  very  various,  both  optically  and 
chemically ;  but  are  all  the  same  in  principle  with  the  method  originally  employed  by 
Talbot.  This  method,  which  was  almost  forgotten  during  the  great  success  of  Daguerre, 
consists  in  first  obtaining,  on  a  transparent  plate,  a  picture  with  lights  and  shades  reversed, 
called  a  negative;  then  placing  this  upon  a  piece  of  paper  sensitized  with  chloride  of  silver, 
and  exposing  it  to  the  sun's  rays.  The  light  parts  of  the  negative  allow  the  light  to  pass 
and  blacken  the  paper,  thus  producing  a  positive  picture.  The  same  negative  serves  for 
producing  a  great  number  of  positives. 

The  negative  plate  is  usually  a  glass  plate  covered  with  a  film  of  collodion  (sometimes 
of  albumen),  sensitized  by  a  salt  of  silver.  The  following  is  one  of  the  numerous  formulae 
for  this  preparation.  Take 

Sulphuric  ether .     300  grammes 

Alcohol  at  40°, .200 

Gun  cotton, 5         „ 

Incorporate  these  ingredients  thoroughly  in  a  porcelain  mortar;  then  add 

Iodide  of  potassium, 13  grammes 

Iodide  of  ammonium, 1'75   „ 

Iodide  of  cadmium, 175   „ 

Bromide  of  cadmium, 1'25   „ 

Thia  mixture  is  poured  over  the  plate,  which  is  then  immersed  in  a  solution  (10  per  cent 


SOLAR   MICROSCOPE.  1029 

1019.  Use  of  Lenses  for  Purposes  of  Projection. — Lenses  are  exten- 
sively employed  in  the  lecture-room,  for  rendering  experiments  visible 
to  a  whole  audience  at  once,  by  projecting  them  on  a  screen.     The 
arrangements  vary  according  to  the  circumstances  of  each  case,  and 
cannot  be  included  in  a  general  description. 

1020.  Solar  Microscope.    Magic  Lantern. — In  the  solar  microscope,  a 
convex  lens  of  short  focal  length  is  employed  to  throw  upon  a  screen 
a  highly-magnified  image  of  a  small  object  placed  a  little  beyond  the 
principal  focus.     As  the  image  is  always  much  less  bright  than  the 


Fig.  732.— Solar  Microscope. 

object,  and  the  more  so  as  the  magnification  is  greater,  it  is  necessary 
that  the  object  should  be  very  highly  illuminated.  For  this  purpose 
the  rays  of  the  sun  are  directed  upon  it  by  means  of  a  mirror  and 
large  lens;  the  latter  serving  to  increase  the  solid  angle  of  the  cone 
of  rays  which  fall  upon  the  object,  and  thus  to  enable  a  larger  por- 
tion of  the  magnifying  lens  to  be  utilized.  The  objects  magnified 
are  always  transparent;  and  the  images  are  formed  by  rays  which 
have  been  transmitted  through  them. 

strong)  of  nitrate  of  silver.  The  film  of  collodion  is  thus  brought  to  an  opal  tint,  and  the 
plate,  after  being  allowed  to  drain,  is  ready  for  exposure  in  the  camera. 

After  being  exposed,  the  picture  is  developed,  by  the  application  of  a  liquid  for  which  the 
following  is  a  formula; 

Distilled  water 250  grammes. 

Pyrogallic  acid, 1         „ 

Crystallizable  acetic  acui, 20         „ 

When  the  picture  is  sufficiently  developed,  it  is  faced,  by  the  application  of  a  solution, 
either  of  hyposulphite  of  soda  from  25  to  30  per  cent  strong,  or  of  cyanide  of  potassium 
3  per  cent  strong,  and  the  negative  is  completed. 

To  obtain  a  positive,  the  negative  plate  is  laid  upon  a  sheet  of  paper  in  a  glass  dish,  the 
paper  having  been  sensitized  by  immersing  it  first  in  a  solution  of  common  sea-salt  3  or  4 
per  cent  strong,  and  then  in  a  solution  of  nitrate  of  silver  18  per  cent  strong.  The  exposure 
is  continued  till  the  tone  is  sufficiently  deep,  the  tint  is  then  improved  by  means  of  a  salt 
of  gold,  and  the  picture  is  fixed  by  hyposulphite  of  soda.  It  has  then  only  to  be  washed 
and  dried. — D. 


1030 


LENSES. 


The  lens  employed  for  producing  the  image  is  usually  compound, 
consisting  of  a  convex  and  a  concave  lens  combined. 

The  electric  light  can  be  employed  instead  of  the  sun.  The 
apparatus  for  regulating  this  light  is  usually  placed  within  a  lantern 
(Fig.  733),  in  such  a  position  that  the  light  is  at  the  centre  of  curva- 


Fig.  733. — Photo-electric  Microscope. 

ture  of  a  spherical  mirror,  so  that  the  inverted  image  of  the  light 
coincides  with  the  light  itself.  The  light  is  concentrated  on  the 
object  by  a  system  of  lenses,  and,  after  passing  through  the  object, 
traverses  another  system  of  lenses,  placed  at  such  a  distance  from  the 
object  as  to  throw  a  highly-magnified  image  of  it  on  a  screen.  The 
whole  arrangement  is  called  the  electric  or  photo-electric  microscope. 
The  magic  lantern  is  a  rougher  instrument  of  the  same  kind, 
employed  for  projecting  magnified  images  of  transparent  paintings, 
executed  on  glass  slides.  It  has  one  lens  for  converging  a  beam  of 
light  on  the  slide,  and  another  for  throwing  an  image  of  the  slide  on 
the  screen.  In  all  these  cases  the  image  is  inverted. 


CHAPTER    LXXI. 


VISION  AND   OPTICAL   INSTRUMENTS. 


1021.  Description  of  the  Eye.— The  human  eye  (Fig.  734)  is  a  nearly 
spherical  ball,  capable  of  turning  in  any  direction  in  its  socket.  Its 
outermost  coat  is  thick  and  horny,  and  is  opaque  except  in  its 


LIVfiuCHEL* 

Fig.  734.— Human  Eye. 

anterior  portion.  Its  opaque  portion  H  is  called  the  sderotica,  or 
in  common  language  the  white  of  the  eye.  Its  transparent  portion  A 
is  called  the  cornea,  and  has  the  shape  of  a  very  convex  watch-glass. 
Behind  the  cornea  is  a  diaphragm  D,  of  annular  form,  called  the  iris. 
It  is  coloured  and  opaque,  and  the  circular  aperture  C  in  its  centre 


1032  VISION   AND    OPTICAL   INSTRUMENTS. 

is  called  the  pupil.  By  the  action  of  the  involuntary  muscles  of  the 
iris,  this  aperture  is  enlarged  or  contracted  on  exposure  to  darkness 
or  light.  The  colour  of  the  iris  is  what  is  referred  to  when  we  speak 
of  the  colour  of  a  person's  eyes.  Behind  the  pupil  is  the  crystalline 
lens  E,  which  has  greater  convexity  at  back  than  in  front.  It  is  built 
up  of  layers  or  shells,  increasing  in  density  inwards,  the  outermost 
shell  having  nearly  the  same  index  of  refraction  as  the  media  in  con- 
tact with  it;  an  arrangement  which  tends  to  prevent  the  loss  of  light 
by  reflection.  The  cavity  B  between  the  cornea  and  the  crystalline  is 
called  the  anterior  chamber,  and  is  filled  with  a  watery  liquid  called 
the  aqueous  humour.  The  much  larger  cavity  L,  behind  the  crystal- 
line, is  called  the  posterior  chamber,  and  is  filled  with  a  transparent 
jelly  called  the  vitreous  humour,  inclosed  in  a  very  thin  transparent 
membrane  (the  hyaloid  membrane).  The  posterior  chamber  is 
inclosed,  except  in  front,  by  the  choroid  coat  or  uvea  I,  which  is 
saturated  with  an  intensely  black  and  opaque  mucus,  called  the 
pigmentum  nigrum.  The  choroid  is  lined,  except  in  its  anterior 
portion,  with  another  membrane  K,  called  the  retina,  which  is 
traversed  by  a  ramified  system  of  nerve  filaments  diverging  from 
the  optic  nerve  M.  Light  incident  on  the  retina  gives  rise  to  the 
sensation  of  vision;  and  there  is  no  other  part  of  the  eye  which 
possesses  this  property. 

1022.  The  Eye  as  an  Optical  Instrument. — It  is  clear,  from  the  above 
description,  that  a  pencil  of  rays  entering  the  eye  from  an  external 
point  will  undergo  a  series  of  refractions,  first  at  the  anterior  surface 
of  the  cornea,  and  afterwards  in  the  successive  layers  of  the  crystal- 
line lens,  all  tending  to  render  them  convergent  (see  table  of  indices, 
§  986).     A  real  and  inverted  image  is  thus  formed  of  any  external 
object  to  which  the  eye  is  directed.   If  this  image  falls  on  the  retina, 
the  object  is  seen;  and  if  the  image  thus  formed  on  the  retina  is 
sharp  and  sufficiently  luminous,  the  object  is  seen  distinctly. 

1023.  Adaptation  to  Different  Distances. — As  the  distance  of  an 
image  from  a  lens  varies  with  the  distance  of  the  object,  it  would 
only  be  possible  to  see  objects  distinctly  at  one  particular  distance, 
were  there  not  special  means  of  adaptation  in  the  eye.     Persons 
whose  sight  is  not  defective  can  see  objects  in  good  definition  at  all 
distances  exceeding  a  certain  limit.     When  we  wish  to  examine  the 
minute  details  of  an  object  to  the  greatest  advantage,  we  hold  it 
at  a  particular  distance,  which  varies  in  different  individuals,  and 
averages  about  eight  inches.     As  we  move  it  further  away,  we 


ADAPTATION   OF   THE   EYE   TO   DISTANCE.  1033 

experience  rather  more  ease  in  looking  at  it,  though  the  diminution 
of  its  apparent  size,  as  measured  by  the  visual  angle,  renders  its 
minuter  features  less  visible.  On  the  other  hand,  when  we  bring  it 
nearer  to  the  eye  than  the  distance  which  gives  the  best  view,  we 
cannot  see  it  distinctly  without  more  or  less  effort  and  sense  of  strain; 
and  when  we  have  brought  it  nearer  than  a  certain  lower  limit 
(averaging  about  six  inches),  we  find  distinct  vision  no  longer  pos- 
sible. In  looking  at  very  distant  objects,  if  our  vision  is  not  defec- 
tive, we  have  very  little  sense  of  effort.  These  phenomena  are  in 
accordance  with  the  theory  of  lenses,  which  shows  that  when  the 
distance  of  an  object  is  a  large  multiple  of  the  focal  length  of  the 
lens,  any  further  increase,  even  up  to  infinity,  scarcely  alters  the 
distance  of  the  image;  but  that,  when  the  object  is  comparatively 
near,  the  effect  of  any  change  of  its  distance  is  considerable.  There 
has  been  much  discussion  among  physiologists  as  to  the  precise  nature 
of  the  changes  by  which  we  adapt  our  eyes  to  distinct  vision  at 
different  distances.  Such  adaptation  might  consist  either  in  a  change 
of  focal  length,  or  in  a  change  of  distance  of  the  retina.  Observa- 
tions in  which  the  eye  of  the  patient  is  made  to  serve  as  a  mirror, 
giving  images  by  reflection  at  the  front  of  the  cornea,  and  at  the 
front  and  back  of  the  crystalline,  have  shown  that  the  convexity  of 
the  front  of  the  crystalline  is  materially  changed  as  the  patient  adapts 
his  eye  to  near  or  remote  vision,  the  convexity  being  greatest  for 
near  vision.  This  increase  of  convexity  corresponds  to  a  shortening 
of  focal  length,  and  is  thus  consistent  with  theory. 

1024.  Binocular  Vision. — The  difficulty  which  some  persons  have 
felt  in  reconciling  the  fact  of  an  inverted  image  on  the  retina  with 
the  perception  of  an  object  in  its  true  position,  is  altogether  fanciful, 
and  arises  from  confused  notions  as  to  the  nature  of  perception. 

The  question  as  to  how  it  is  that  we  see  objects  single  with  two 
eyes,  rests  upon  a  different  footing,  and  is  not  to  be  altogether  ex- 
plained by  habit  and  association.1  To  each  point  in  the  retina  of  one 
eye  there  is  a  corresponding  point,  similarly  situated,  in  the  other. 
An'  impression  produced  on  one  of  these  points  is,  in  ordinary  cir- 
cumstances, undistinguishable  from  a  similar  impression  produced  on 
the  other,  and  when  both  at  once  are  similarly  impressed,  the  effect 
is  simply  more  intense  than  if  one  were  impressed  alone;  or,  to 
describe  the  same  phenomena  subjectively,  we  have  only  one  field 

1  Binocular  vision  is  a  subject  which  has  been  much  debated.  For  the  account  here 
given  of  it,  the  Editor  is  responsible. 


1034  VISION  AND   OPTICAL  INSTRUMENTS. 

of  view  for  our  two  eyes,  and  in  any  part  of  this  field  of  view  we  see 
either  one  image,  brighter  than  we  should  see  it  by  one  alone,  or  else 
we  see  two  overlapping  images.  This  latter  phenomenon  can  be 
readily  illustrated  by  holding  up  a  finger  between  one's  eyes  and  a 
wall,  and  looking  at  the  wall.  We  shall  see,  as  it  were,  two  trans- 
parent fingers  projected  on  the  wall.  One  of  these  transparent 
fingers  is  in  fact  seen  by  the  right  eye,  and  the  other  by  the  left,  but 
our  visual  sensations  do  not  directly  inform  us  which  of  them  is  seen 
by  the  right  eye,  and  which  by  the  left. 

The  principal  advantage  of  having  two  eyes  is  in  the  estimation 
of  distance,  and  the  perception  of  relief.  In  order  to  see  a  point  as 
single  by  two  eyes,  we  must  make  its  two  images  fall  on  correspond- 
ing points  of  the  retinae;  and  this  implies  a  greater  or  less  converg- 
ence of  the  optic  axes  according  as  the  object  is  nearer  or  more  remote. 
We  are  thus  furnished  with  a  direct  indication  of  the  distance  of  the 
object  from  our  eyes;  and  this  indication  is  much  more  precise  than 
that  derived  from  the  adjustment  of  their  focal  length. 

In  judging  of  the  comparative  distances  of  two  points  which  lie 
nearly  in  the  same  direction,  we  are  greatly  aided  by  the  parallactic 
displacement  which  occurs  when  we  change  our  own  position. 

We  can  also  form  an  estimate  of  the  nearness  of  an  object,  from 
the  amount  of  change  in  its  apparent  size,  contour,  and  bearing,  pro- 
duced by  shifting  our  position.  This  would  seem  to  be  the  readiest 
means  by  which  very  young  animals  can  distinguish  near  from 
remote  objects. 

1025.  Stereoscope. — The  perception  of  relief  is  closely  connected 
with  the  doubleness  of  vision  which  occurs  when  the  images  on 
corresponding  portions  of  the  two  retinaa  are  not  similar.  In  survey- 
ing an  object  we  run  our  eyes  rapidly  over  its  surface,  in  such  a  way 
as  always  to  attain  single  vision  of  the  particular  point  to  which  our 
attention  is  for  the  instant  directed.  We  at  the  same  time  receive 
a  somewhat  indistinct  impression  of  all  the  points  within  our  field 
of  view;  an  impression  which,  when  carefully  analysed,  is  found  to 
involve  a  large  amount  of  doubleness.  These  various  impressions 
combine  to  give  us  the  perception  of  relief;  that  is  to  say,  of  form 
in  three  dimensions. 

The  perception  of  relief  in  binocular  vision  is  admirably  illustrated 
by  the  stereoscope,  an  instrument  which  was  invented  by  Wheatstone, 
and  reduced  to  its  present  more  convenient  form  by  Brewster.  Two 
figures  are  drawn,  as  in  Fig.  735,  being  perspective  representations  of 


STEREOSCOPE. 


1035 


the  same  object  from  two  neighbouring  points  of  view,  such  as  might 
be  occupied  by  the  two  eyes  in  looking  at  the  object.  Thus  if  the 
object  be  a  cube,  the  right  eye  will  have  a  fuller  view  of  the  right 


Fig.  735. -Stereoscopic  Pictures. 


Fig.  736.— Stereoscope. 


Fig.  737.— Path  of  Rays  in  Stereoscope. 


face,  and  the  left  eye  of  the  left  face.  The  two  pictures  are  placed 
in  the  right  and  left  compartments  of  a  box,  which  has  a  partition 
down  the  centre  serving  to  insure  that  each  eye  shall  see  only  the 
picture  intended  for  it;  and  over  each  of  the  compartments  a  half- 
lens  is  fixed,  serving,  as  in  Fig.  737,  not  only  to  magnify  the  picture, 
but  at  the  same  time  to  displace  it,  so  that  the  two  virtual  images 
are  brought  into  approximate  coincidence.  Stereoscopic  pictures 
are  usually  photographs  obtained  by  means  of  a  double  camera, 
having  two  objectives,  one  beside  the  other,  which  play  the  part 
of  two  eyes. 

When  matters  are  properly  arranged,  the  observer  seems  to  see 
the  object  in  relief.  He  finds  himself  able  to  obtain  single  view  of 
any  one  point  of  the  solid  image  which  is  before  him;  and  the  adjust- 
ments of  the  optic  axes  which  he  finds  it  necessary  to  make,  in  shift- 
ing his  view  from  one  point  of  it  to  another,  are  exactly  such  as 
would  be  required  in  looking  at  a  solid  object. 

When  one  compartment  of  the  stereoscope  is  empty,  and  the  other 
contains  an  object,  an  observer,  of  normal  vision,  looking  in  in  the 
ordinary  way,  is  unable  to  say  which  eye  sees  the  object.  If-  two 
pictures  are  combined,  consisting  of  two  equal  circles,  one  of  them 


1036  VISION   AND   OPTICAL   INSTRUMENTS. 

having  a  cross  in  its  centre,  and  the  other  not,  he  is  unable  to  decide 
whether  he  sees  the  cross  with  one  eye  or  both. 

When  two  entirely  dissimilar  pictures  are  placed  in  the  two  com- 
partments, they  compete  for  mastery,  each  of  them  in  turn  becoming 
more  conspicuous  than  the  other,  in  spite  of  any  efforts  which  the 
observer  may  make  to  the  contrary.  A  similar  fluctuation  will  be 
observed  on  looking  steadily  at  a  real  object  which  is  partially  hidden 
from  one  eye  by  an  intervening  object.  This  tendency  to  alternate 
preponderance  renders  it  well  nigh  impossible  to  combine  two  colours 
by  placing  one  under  each  eye  in  the  stereoscope. 

The  immediate  visual  impression,  when  we  look  either  at  a  real 
solid  object,  or  at  the  apparently  solid  object  formed  by  properly 
combining  a  pair  of  stereoscopic  views,  is  a  single  picture  formed  of 
two  slightly  different  pictures  superimposed  upon  each  other.  The 
coincidence  becomes  exact  at  any  point  to  which  attention  is  directed, 
and  to  which  the  optic  axes  are  accordingly  made  to  converge,  but  in 
the  greater  part  of  the  combined  picture  there  is  a  want  of  coin- 
cidence, which  can  easily  be  detected  by  a  collateral  exercise  of 
attention.  The  fluctuation  above  described  to  some  extent  tends  to 
conceal  this  doubleness;  and  in  looking  at  a  real  solid  object,  the 
concealment  is  further  assisted  by  the  blurring  of  parts  which  are 
out  of  focus. 

1026.  Visual  Angle.  Magnifying  Power. — The  angle  which  a  given 
straight  line  subtends  at  the  eye  is  called  its  visual  angle,  or  the 
angle  under  which  it  is  seen.  This  angle  is  the  measure  of  the 
length  of  the  image  of  the  straight  line  on  the  retina.  Two  discs 
at  different  distances  from  the  eye,  are  said  to  have  the  same  ap- 
parent size,  if  their  diameters  are  seen  under  equal  angles.  This  is 
the  condition  that  the  nearer  disc,  if  interposed  between  the  eye 
and  the  remoter  disc,  should  be  just  large  enough  to  conceal  it  from 
view. 

The  angle  under  which  a  given  line  is  seen,  evidently  depends  not 
only  on  its  real  length,  and  the  direction  in  which  it  points,  but  also 
on  its  distance  from  the  eye;  and  varies,  in  the  case  of  small  visual 
angles,  in  the  inverse  ratio  of  this  distance.  The  apparent  length  of 
a  straight  line  may  be  regarded  as  measured  by  the  visual  angle 
which  it  subtends. 

By  the  magnifying  power  of  an  optical  instrument,  is  usually 
meant  the  ratio  in  which  it  increases  apparent  lengths  in  this  sense. 
In  the  case  of  telescopes,  the  comparison  is  between  an  object  as 


MAGNIFICATION.  1037 

seen  in  the  telescope,  and  the  same  object  as  seen  with  the  naked 
eye  at  its  actual  distance.  In  the  case  of  microscopes,  the  compari- 
son is  between  the  object  as  seen  in  the  instrument,  and  the  same 
object  as  seen  by  the  naked  eye  at  the  least  distance  of  distinct 
vision,  which  is  usually  assumed  as  10  inches. 

But  two  discs,  whose  diameters  subtend  the  same  angle  at  the  eye, 
may  be  said  to  have  the  same  apparent  area;  and  since  the  areas 
of  similar  figures  are  as  the  squares  of  their  linear  dimensions,  it  is 
evident  that  the  apparent  area  of  an  object  varies  as  the  square  of 
the  visual  angle  subtended  by  its  diameter.  The  number  expressing 
magnification  of  apparent  area  is  therefore  the  square  of  the  mag- 
nifying power  as  above  defined.  Frequently,  in  order  to  show  that 
the  comparison  is  not  between  apparent  areas,  but  between  apparent 
lengths,  an  instrument  is  said  to  magnify  so  many  'diameters.  If 
the  diameter  of  a  sphere  subtends  1°  as  seen  by  the  naked  eye,  and 
10°  as  seen  in  a  telescope,  the  telescope  is  said  to  have  a  magnifying 
power  of  10  diameters.  The  superficial  magnification  in  this  case  is 
evidently  100. 

The  apparent  length  and  apparent  area  of  an  object  are  respect- 
ively proportional  to  the  length  and  area  of  its  image  on  the 
retina. 

Apparent  length  is  measured  by  the  plane  angle,  and  apparent 
area  by  the  solid  angle,  which  an  object  subtends  at  the  eye. 

1027.  Spectacles. — Spectacles  are  of  two  kinds,  intended  to  remedy 
two  opposite  defects  of  vision.  Short-sighted  persons  can  see  objects 
distinctly  at  a  smaller  distance  than  persons  whose  vision  is  normal; 
but  always  see  distant  objects  confused.  On  the  other  hand,  persons 
whose  vision  is  normal  in  their  youth,  usually  become  over-sighted 
with  advancing  years,  so  that,  while  they  can  still  adjust  their  eyes 
correctly  for  distant  vision,  objects  as  near  as  10  or  12  inches  always 
appear  blurred.  Spectacles  for  over-sighted  persons  are  convex,  and 
should  be  of  such  focal  length,  that,  when  an  object  is  held  at  about 
10  inches  distance,  its  virtual  image  is  formed  at  the  nearest  distance 
of  distinct  vision  for  the  person  who  is  to  use  them.  This  latter 
distance  must  be  ascertained  by  trial.  Call  it  p  inches;  then,  by 
§  1012,  the  formula  for  computing  the  required  focal  length  x  (in 
inches)  is 

!    i_i 

10      p  ~~  x' 

For  example,  if  15  inches  is  the  nearest  distance  at  which  the  person 


1038 


VISION  AND   OPTICAL   INSTRUMENTS. 


can  conveniently  read  without  spectacles,  the  focal  length  required 
is  30  inches. 

In  Fig.  738,  A  represents  the  position  of  a  small  object,  and  A' 
that  of  its  virtual  image  as  seen  with  spectacles  of  this  kind. 

Over-sight  is  not  the  only  defect  which  the  eye  is  liable  to  acquire 


Fig.  738.— Spectacle-glass  for  Over-sighted  Eye. 

by  age,  but  it  is  the  defect  which  ordinary  spectacles  are  designed 
to  remedy. 

Spectacles  for  short-sighted  persons  are  concave,  and  the  focal 


Fig.  739.— Spectacle-glass  for  Short-sighted  Eye. 


length  which  they  ought  to  have,  if  designed  for  reading,  may  be 
computed  by  the  formula 


p  denoting  the  nearest  distance  at  which  the  person  can  read,  and  x 
the  focal  length,  both  in  inches.  If  his  greatest  distance  of  distinct 
vision  exceeds  the  focal  length,  he  will  be  able,  by  means  of  the 
spectacles,  to  obtain  distinct  vision  of  objects  at  all  distances,  from 
10  inches  upwards. 
1028.  Simple  Magnifier. — A  magnifying  glass  is  a  convex  lens,  of 


MAGNIFYING  GLASS.  1039 

shorter  focal  length  than  the  human  eye,  and  is  placed  at  a  distance 
somewhat  less  than  its  focal  length  from  the  object  to  be  viewed. 
In  Fig.  740,  a  b  is  the 
object,  and  A  B  the 
virtual  image  which 
is  seen  by  the  eye 
K.  The  construction 
which  we  have  em- 
ployed for  drawing 

the      image      is      One  Fig.  740.— Magnifying  Glass. 

which  we  have  seve- 
ral times  used  before.     Through  the  point  a,  the  line  a  M  is  drawn 
parallel  to  the  principal  axis.    F  M  is  then  drawn  from  the  principal 
focus  F;  O  a  is  drawn  from  the  optical  centre  O;  and  these  two  lines 
are  produced  till  they  meet  in  A. 

Distance  of  lens  from  object.  In  order  that  the  image  may  be 
properly  seen,  its  distance  from  the  eye  must  fall  between  the  limits 
of  distinct  vision;  and  in  order  that  it  may  be  seen  under  the  largest 
possible  visual  angle,  the  eye  must  be  close  to  the  lens,  and  the 
object  must  be  as  near  as  is  compatible  with  distinct  vision.  This 
and  other  interesting  properties  are  established  by  the  following 
investigation: — 

Let  B  denote  the  visual  angle  under  which  the  observer  sees  the 
image  of  the  portion  a  c  of  the  object.  Also  let  x  denote  the  distance 
c  O  of  the  object  from  the  lens,  and  y  the  distance  O  K  of  the  lens 
from  the  eye.  Then  we  have 

tang-  AC          AC    ; 
CK  ~  CO  +  y* 

but,  by  formulae  (10)  and  (11)  of  last  chapter,  we  have 


Substituting  these  values  for  A  C  and  C  0,  and  reducing,  we  have 

(A) 

This  equation  shows  that,  for  a  given  lens  and  a  given  object,  the 
visual  angle  varies  inversely  as  the  quantity  (%+y)  f-xy- 


1040  VISION  AND   OPTICAL  INSTRUMENTS. 

The  following  practical  consequences  are  easily  drawn: — 

(1)  If  the  distance  x  +  y  of  the  eye  from  the  object  is  given,  the 
visual  angle  increases  as  the  two  distances  x,  y  approach  equality, 
and  is  not  altered  by  interchanging  them. 

(2)  If  one  of  the  two  distances  x,  y  be  given,  and  be  less  than  /, 
the  other  must  be  made  as  small  as  possible,  if  we  wish  to  obtain 
the  largest  possible  visual  angle. 

To  obtain  the  absolute  maximum  of  visual  angle,  we  must  select, 
from  the  various  positions  which  make  C  K  equal  to  the  nearest 
distance  of  distinct  vision,  that  which  gives  the  largest  value  of  A  C, 
since  the  quotient  of  A  C  by  C  K  is  the  tangent  of  the  visual  angle. 
Now  A  C  increases  as  the  image  moves  further  from  the  lens,  and 
hence  the  absolute  maximum  is  obtained  by  making  its  distance 
from  the  lens  equal  to  the  nearest  distance  of  distinct  vision,  and 
making  the  eye  come  up  close  to  the  lens.  In  this  case  the  distanc0 
p  of  the  object  from  the  lens  is  given  by  the  equation  —  5  ~  7' 
where  D  denotes  the  nearest  distance  of  distinct  vision;  and  tan  6  is 
—  or  ac  (7  +  p  J.  But  the  greatest  angle  under  which  the  body  could 
be  seen  by  the  naked  eye  is  the  angle  whose  tangent  is  ^;  hence  the 
visual  angle  (or  its  tangent)  is  increased  by  the  lens  in  the  ratio 
1+  ,,  which  is  called  the  'magnifying  power.  If  the  object  were  in 
the  principal  focus,  and  the  eye  close  to 
the  lens,  the  magnifying  power  would  be  ->• 

In  either  case,  the  thickness  of  the  lens 
being  neglected,  the  visual  angle  is  the  angle 
which  the  object  subtends  at  the  centre  of 
the  lens,  and  therefore  varies  inversely  as 
the  distance  of  this  centre  from  the  object. 
For  lenses  of  small  focal  length,  the  recip- 
rocal of  the  focal  length  may  be  regarded 
as  proportional  to  the  magnifying  power. 

Simple  Microscope. — By  a  simple  micro- 
fig.  T4i.-stapie  Microscope,      scope  is  usually  understood  a  lens  of  short 
focal  length  mounted  in  a  manner  convenient 

for  the  examination  of  small  objects.  Fig.  741  represents  an  instru- 
ment of  this  kind.  The  lens  I  is  mounted  in  brass,  and  carried  at  the 
end  of  an  arm.  It  is  raised  and  lowered  by  turning  the  milled  head  V, 


MICROSCOPE.  1041 

which  acts  on  the  rack  a.  C  is  the  platform  on  which  the  object  is 
laid,  and  M  is  a  concave  mirror,  which  can  be  employed  for  increas- 
ing the  illumination  of  the  object. 

1029.  Compound  Microscope. — In  the 
compound  microscope,  there  is  one  lens 
which  forms  a  real  and  greatly  enlarged 
image  of  the  object;  and  this  image  is 
itself  magnified  by  viewing  it  through 
another  lens 

In  Fig.  742,  a  b  is  the  object,  O  is  the 
first  lens,  called  the  objective,  and  is 
placed  at  a  distance  only  slightly  ex- 
ceeding its  focal  length  from  the  object; 
an  inverted  image  %  b1  is  thus  formed 
at  a  much  greater  distance  on  the  other 
side  of  the  lens, and  proportionally  larger. 
O'  is  the  second  lens,  called  the  ocular 
or  eye-piece,  which  is  placed  at  a  dis- 
tance a  little  less  than  its  focal  length 
from  the  first  image  a1  6P  and  thus  forms 
an  enlarged  virtual  ima^e  of  it  A  B,  at 

*?      .    ,.   .  -        «   ..  Fig.  742.-Compound  Microscope. 

a  convenient  distance  for  distinct  vision. 

If  we  suppose  the  final  image  A  B  to  be  at  the  least  distance  of 
distinct  vision  from  the  eye  placed  at  O'  (this  being  the  arrangement 
which  gives  the  largest  visual  angle),  the  magnifying  power  will  be 
simply  the  ratio  of  the  length  of  this  image  to  that  of  the  object  a  b, 
and  will  be  the  product  of  the  two  factors  ^-?  and  — \l.  The  former 

tti  D!  (to 

is  the  magnification  produced  by  the  eye-piece,  and  is,  as  we  have 
just  shown  (§  1028),  1  -f  ->.  The  other  factor  °^y  is  the  magnification 
produced  by  the  objective,  and  is  equal  to  the  ratio  of  the  distances 
?"'.  If  the  objective  is  taken  out,  and  replaced  by  another  of 
different  focal  length,  the  readjustment  will  consist  in  altering  the 
distance  0  a,  leaving  the  distance  O  al  unchanged.  The  total  magni- 
fication therefore  varies  inversely  as  O  a,  that  is,  nearly  in  the  inverse 
ratio  of  the  focal  length  of  the  objective.  Compound  microscopes 
are  usually  provided  with  several  objectives,  of  various  focal  lengths, 
from  which  the  observer  makes  a  selection  according  to  the  magni- 
fying power  which  he  requires  for  the  object  to  be  examined.  The 
powers  most  used  range  from  50  to  350  diameters. 
66 


1042  VISION   AND   OPTICAL   INSTRUMENTS. 

The  magnifying  power  of  a  microscope  can  be  determined  by 
direct  observation,  in  the  following  way.  A  plane  reflector  pierced 
with  a  hole  in  its  centre,  is  placed  directly  over  the  eye-piece  (Fig. 
743),  at  an  inclination  of  45°,  and  an- 
other plane  reflector,  or  still  better,  a 
totally  reflecting  prism,  as  in  the  figure, 
is  placed  parallel  to  it  at  the  distance  of 
an  inch  or  two,  so  that  the  eye,  looking 
down  upon  the  first  mirror,  sees,  by  means 
of  two  successive  reflections,  the  image 
of  a  divided  scale  placed  at  a  distance  of 
8  or  10  inches  below  the  second  reflector. 
In  taking  an  observation,  a  micrometer 
scale  engraved  on  glass,  its  divisions  being 
at  a  known  distance  apart  (say  3-^-5-  of  a 
millimetre),  is  placed  in  the  microscope  as 
the  object  to  be  magnified;  and  the  ob- 
server holds  his  eye  in  such  a  position 
that,  by  means  of  different  parts  of  his 
pupil,  he  sees  at  once  the  magnified  image 
of  the  micrometer  scale  in  the  microscope, 
and  the  reflected  and  unmagnified  image 
•  743-  of  the  other  scale.  The  two  images  will 

Measurement  of  Magnifying  Power.  .  .  °       . 

be  superimposed  in  the  same  field  of  view; 

and  it  is  easy  to  observe  how  many  divisions  of  the  one  coincide 
with  a  given  number  of  divisions  of  the  other.  Let  the  divisions  on 
the  large  scale  be  millimetres,  and  those  on  the  micrometer  scale 
hundredths  of  a  millimetre.  Then  the  magnifying  power  is  100,  if 
one  of  the  magnified  covers  one  of  the  unmagnified  divisions;  and 

is  ~— ,  if  n  of  the  former  cover  N  of  the  latter.     This  is  on  the 

assumption  that  the  large  scale  is  placed  at  the  nearest  distance  of 
convenient  vision.  In  stating  the  magnifying  power  of  a  microscope, 
this  distance  is  usually  reckoned  as  10  inches. 

A  short-sighted  person  sees  an  image  in  a  microscope  (whether 
simple  or  compound)  under  a  larger  visual  angle  than  a  person  of 
normal  sight;  but  the  inequality  is  not  so  great  as  in  the  case  of 
objects  seen  by  the  naked  eye.  In  fact,  if  /  be  the  focal  length  of 
the  eye-piece  in  a  compound  microscope,  or  of  the  microscope  itself 
if  simple,  and  D  the  nearest  distance  of  distinct  vision  for  the 


TELESCOPE.  1043 

observer,  the  visual  angle  under  which  the  image  is  seen  in  the 
microscope  is  proportional  to  j  +  ^,  the  greatest  visual  angle  for  the 

naked  eye  being  represented  by  jy     Both  these  angles  increase  as 

D  diminishes,  but  the  latter  increases  in  a  greater  ratio  than  the 
former.  When  /  is  as  small  as  TV  of  an  inch,  the  visual  angle  in 
the  microscope  is  sensibly  the  same  for  short  as  for  normal  sight. 

Before  reading  off  the  divisions  in  the  observation  above  described; 
care  should  be  taken  to  focus  the  microscope  in  such  a  way,  that  the 
image  of  the  micrometer  scale  is  at  the  same  distance  from  the  eye 
as  the  image  of  the  large  scale  with  which  it  is  compared.  When 
this  is  done,  a  slight  motion  of  the  eye  does  not  displace  one  image 
with  respect  to  the  other. 

Instead  of  a  single  eye-lens,  it  is  usual  to  employ  two  lenses 
separated  by  an  interval,  that  which  is  next  the  eye  being  called 
the  eye-glass,  and  the  other  the  field-glass.  This  combination  is 
equivalent  to  the  Huygenian  or  negative  eye-piece  employed  in 
telescopes  (§  1070). 

1030.  Astronomical  Telescope. — The  astronomical  refracting  tele- 
scope consists  essen- 
tially (like  the  com- 
pound microscope)  of 
two  lenses,  one  of 
which  forms  a  real  and 
inverted  image  of  the 
object, which  is  looked 
at  through  the  other. 

In  Fig.  744, 0  is  the 

object-glass,  which   is  Fig.  TM. -Astronomical  Telescope. 

sometimes  a  foot   or 

more  in  diameter,  and  is  always  of  much  greater  focal  length  than 
the  eye-piece  O'.  The  inverted  image  of  a  distant  object  is  formed 
at  tile  principal  focus  F.  This  image  is  represented  at  a  6,  The 
parallel  rays  marked  1,  2  come  from  the  upper  extremity  of  the 
object,  and  meet  at  a;  and  the  parallel  ray»  3,  4,  from  the  other 
extremity,  meet  at  6.  A'  B'  is  the  virtual  image  of  a  6  formed  by  the 
eye-piece.  Its  distance  from  the  eye  can  be  changed  by  pulling  out 
or  pushing  in  the  eye-tube;  and  may  in  practice  have  any  value 
intermediate  between  the  least  distance  of  distinct  vision  and  infinity, 
the  visual  angle  under  which  it  is  seen  being  but  slightly  affected  by 


1044 


VISION    AND    OPTICAL   INSTRUMENTS. 


this  adjustment.  The  rays  from  the  highest  point  of  the  object 
emerge  from  the  eye-piece  as  a  pencil  diverging  from  A';  and  the  rays 
from  the  lowest  point  of  the  object  form  a  pencil  diverging  from  B'. 
Magnification. — The  angle  under  which  the  object  would  be  seen 
by  the  naked  eye  is  a  O  6;  for  the  rays  a  0, 6  0,  if  produced,  would 
pass  through  its  extremities.  The  angle  under  which  it  is  seen  in 
the  telescope,  if  the  eye  be  close  to  the  eye-lens,  is  A'  O'  B'  or  a  0'  b. 

The  magnification  is  therefore  |^,  which  is  approximately  the 
same  as  the  ratio  of  the  distances  of  the  image  a  b  from  the  two  lenses 
^.  If  the  eye-tube  is  so  adjusted  as  to  throw  the  image  A'  B'  to 

infinite  distance,  F  will  be  the  principal  focus  of  both  lenses,  and 
the  magnification  is  the  ratio  of  the  focal  length  of  the  object-glass 

to  that  of  the  eye-piece. 
If  the  eye -tube  be 
pushed  in  as  far  as  is 
compatible  with  dis- 
tinct vision  (the  eye 
being  close  to  the  lens), 
the  magnification  is 
greater  than  this  in  the 

ratio  -f&,  D  denoting 
the  nearest  distance  of 
distinct  vision,  and  / 
the  focal  length  of  the 
eye-piece 

The  magnification 
can  be  directly  ob- 
served by  looking  with  one  eye  through  the  telescope  at  a  brick 
wall,  while  the  other  eye  is  kept  open.  The  image  will  thus  be 
superimposed  on  the  actual  wall,  and  we  have  only  to  observe  how 
many  courses  of  the  latter  coincide  with  a  single  course  of  the 
magnified  image. 

If  the  telescope  is  large,  its  tube  may  prevent  the  second  eye  from 
seeing  the  wall,  and  it  may  be  necessary  to  employ  a  reflecting 
arrangement,  as  in  Fig.  745,  analogous  to  that  described  in  connec- 
tion with  the  microscope. 

Telescopes  without  stands  seldom  magnify  more  than  about  10 
diameters.  Powers  of  from  20  to  60  are  common  in  telescopes  with 


Fig.  745.— Measurement  of  Magnifying  Power. 


THE  BRIGHT   SPOT. 


1045 


Fig.  746.— Astronomical  Telescope. 


stands,  intended  for  terrestrial  purposes.  The  powers  chiefly  em- 
ployed in  astronomical  observation  are  from  100  to  500. 

Mechanical  Arrangements. — The  achromatic  object-glass  0  is  set 
in  a  mounting  which  is  screwed  into  one  end  of  a  strong  brass  tube 
A  A  (Fig.  746).  In  the 
other  end  slides  a  smaller 
tube  F  containing  the  eye- 
piece O';  and  by  turning 
the  milled  head  V  in  one 
direction  or  the  other,  the 
eye -piece  is  moved  for- 
wards or  backwards. 

Finder.  —  The  small 
telescope  I,  which  is  at- 
tached to  the  principal 
telescope,  is  called  a  finder. 
This  appendage  is  indis- 
pensable when  the  prin- 
cipal .  telescope  has  a  high 
magnifying  power;  for  a 

high  magnifying  power  involves  a  small  field  of  view,  and  consequent 
difficulty  in  directing  the  telescope  so  as  to  include  a  selected  object 
within  its  range.  The  finder  is  a  telescope  of  large  field;  and  as  it 
is  set  parallel  to  the  principal  telescope,  objects  will  be  visible  in 
the  latter  if  they  are  seen  in  the  centre  of  the  field  of  view  of  the 
former. 

1031.  Best  Position  for  the  Eye. — The  eye-piece  forms  a  real  and 
inverted  image  of  the  object-glass1  at  E  E'  (Fig.  744),  through  which 
all  rays  transmitted  by  the  telescope  must  of  necessity  pass.  If  the 
telescope  be  directed  to  a  bright  sky,  and  a  piece  of  white  paper  held 
behind  the  eye-piece  to  serve  as  a  screen,  a  circular  spot  of  light  will 
be  formed  upon  it,  which  will  become  sharply  defined  (and  at  the 
same  time  attain  its  smallest  size)  when  the  screen  is  held  in  the 
correct  position.  This  image  (which  we  shall  call  the  bright  spot) 
may  be  regarded  as  marking  the  proper  place  for  the  pupil  of  the 
observer's  eye.  Every  ray  which  traverses  the  centre  of  the  object- 
glass  traverses  the  centre  of  this  spot;  every  ray  which  traverses 
the  upper  edge  of  the  object-glass  traverses  the  lower  edge  of  the 

1  Or  it  may  be  called  an  image  of  the  aperture  which  the  object-gluts  Jills.  It  remains  sen- 
sibly unchanged  on  removing  the  object-glass  so  as  to  leave  the  end  of  the  telescope  open. 


1046  VISION   AND   OPTICAL  INSTRUMENTS. 

spot;  and  any  selected  point  of  the  spot  receives  all  the  rays  which 
have  been  transmitted  by  one  particular  point  of  the  object-glass. 
An  eye  with  its  pupil  anywhere  within  the  limits  of  the  bright  spot, 
will  therefore  see  the  whole  field  of  view  of  the  telescope.  If  the 
spot  and  pupil  are  of  exactly  the  same  size,  they  must  be  made  to 
coincide  with  one  another,  as  the  necessary  condition  of  seeing  the 
whole  field  of  view  with  the  brightest  possible  illumination.  Usually 
in  practice  the  spot  is  much  smaller  than  the  pupil,  so  that  these 
advantages  can  be  obtained  without  any  nicety  of  adjustment;  but 
to  obtain  the  most  distinct  vision,  the  centre  of  the  pupil  should 
coincide  as  closely  as  possible  with  the  centre  of  the  spot.  To  facili- 
tate this  adjustment,  a  brass  diaphragm,  with  a  hole  in  its  centre, 
is  screwed  into  the  eye-end  of  the  telescope,  the  proper  place  for  the 
eye  being  close  to  this  hole. 

One  method  of  determining  the  magnifying  power  of  a  telescope 
consists  in  measuring  the  diameter  of  the  bright  spot,  and  comparing 
it  with  the  effective  aperture  of  the  object-glass.  In  fact,  let  F  and 
/  denote  the  focal  lengths  of  object-glass  and  eye-piece,  and  a  the 
distance  of  the  spot  from  the  centre  of  the  eye-piece;  then  F+/  is 
approximately  the  distance  of  the  object-glass  from  the  same  centre, 

and,  by  the  formula  for  conjugate  focal  distances,  we  have  „—  -«  +  —  =  y' 
Multiplying  both  sides  of  this  equation  by  F+/,  and  then  subtract- 


ing unity,  we  have  ~=j-  But  the  ratio  of  the  diameter  of  the 
object-glass  to  that  of  its  image  is  -^;  and  ->  is  the  usual  formula 

for  the  magnifying  power.  Hence,  the  linear  magnifying  power  of 
a  telescope  is  the  ratio  of  the  diameter  of  the  object-glass  to  that  of 
the  bright  spot. 

1032.  Terrestrial  Telescope.  —  The  astronomical  telescope  just  de- 
scribed gives  inverted  images.  This  is  no  drawback  in  astronomical 
observation,  but  would  be  inconvenient  in  viewing  terrestrial  objects. 
In  order  to  re-invert  the  image,  and  thus  make  it  erect,  two  addi- 
tional lenses  O"O'"  (Fig.  747)  are  introduced  between  the  real 
image  a  b  and  the  eye-lens  O'.  If  the  first  of  these  0"  is  at  the  dis- 
tance of  its  principal  focal  length  from  a  b,  the  pencils  which  fall 
upon  the  second  will  be  parallel,  and  an  erect  image  a  b'  will  thus 
be  formed  in  the  principal  focus  of  O'".  This  image  is  viewed  through 
the  eye-lens  O',  and  the  virtual  image  A'  B'  which  is  perceived  by 
the  eye  will  therefore  be  erect.  The  two  lenses  0",  0'",  are  usually 


GALILEAN   TELESCOPE. 


1047 


made  precisely  alike,  in  which  case  the  two  images  a  b,  a  V  will  be 
equal.     In  the  better  class  of  terrestrial  telescopes,  a  different  ar- 


Fig.  747.-Terrestrial  Eye-piece. 

rangement  is  adopted, requiring  one  more  lens;  but  whatever  system 
be  employed,  the  reinversion  of  the  image  always  involves  some  loss 
both  of  light  and  of  distinctness. 

1033.  Galilean  Telescope. — Besides  the  disadvantages  just  men- 
tioned, the  erecting  eye-piece  involves  a  considerable  addition  to  the 
length  of  the  instrument.     The  telescope  invented  by  Galileo,  and 
the  earliest  of  all  tele- 
scopes, gives  erect  im- 
ages with  only  two  len- 
ses, and  with  shorter 
length  than  even  the 
astronomical  telescope. 
0  (Fig.  748)  is  the  ob- 
ject-glass, which  is  con- 
vex as  in  the  astrono- 
mical   telescope,    and 
would  form  a  real  and 
inverted  image  a  b  at 

its  principal  focus;  but  the  eye-glass  O',  which  is  a  concave  lens, 
is  interposed  at  a  distance  equal  to  or  slightly  exceeding  its  own 
focal  length  from  the  place  of  this  image,  and  forms  an  erect  virtual 
image  A'  B',  which  the  observer  sees. 

Neglecting  the  distance  of  his  eye  from  the  lens,  the  angle  under 
which  he  sees  the  image  is  A'  O'  B',  which  is  equal  to  a  0  b,  whereas 


Fig.  748.  -Galilean  Telescope. 


1048  VISION    AND    OPTICAL   INSTRUMENTS. 

the  visual  angle  to  the  naked  eye  would  be  a  O  b.  The  magnification 
is  therefore  ^|,  which  is  approximately  equal  to  ~^,  c  being  the 

principal  focus  of  the  object-glass.  If  the  instrument  is  focussed  in 
such  a  way  that  the  image  A'  B'  is  thrown  to  infinite  distance,  c  is 
also  the  principal  focus  of  the  eye-lens,  and  the  magnification  is 
simply  the  ratio  of  the  focal  lengths  of  the  two  lenses.  This  is 
the  same  rule  which  we  deduced  for  the  astronomical  telescope; 
but  the  Galilean  telescope,  if  of  the  same  power,  is  shorter  by 
twice  the  focal  length  of  the  eye-lens,  since  the  distance  between 
the  two  lenses  is  the  difference  instead  of  the  sum  of  their  focal 
lengths. 

This  telescope  has  the  disadvantage  of  not  admitting  of  the  em- 
ployment of  cross  wires;  for  these,  in  order  to  serve  their  purpose, 
must  coincide  with  the  real  image;  and  no  such  image  exists  in 
this  telescope. 

There  is  another  peculiarity  in  the  absence  of 
the  bright  spot  above  described,  the  image  of  the 
object-glass  formed  by  the  eye-glass  being  virtual. 
In  other  telescopes,  if  half  the  object-glass  be  cov- 
ered, half  the  bright  spot  will  be  obliterated;  but 
the  remaining  half  suffices  for  giving  the  whole 

LHi  field  of  view,  though  with  diminished  brightness. 
I  In  the  Galilean  telescope,  on  the  contrary,  if  half 
I  the  object-glass  be  covered,  half  the  field  of  view 
B»  will  be  cut  off,  and  the  remaining  half  will  be 
unaffected. 


Fig.  749. -opera-glass.  The  opera-glass,  single  or  binocular,  is  a  Gali- 
lean telescope,  or  a  pair  of  Galilean  telescopes.  In 
the  best  instruments,  both  object-glass  and  eye-glass  are  achromatic 
combinations  of  three  pieces,  as  shown  in  section  in  the  figure  (Fig. 
749) ;  the  middle  piece  in  each  case  being  flint,  and  the  other  two 
crown  (§  1064). 

1034.  Reflecting  Telescopes. — In  reflecting  telescopes,  the  place  of 
an  object-glass  is  supplied  by  a  concave  mirror  called  a  speculum, 
usually  composed  of  solid  metal.  The  real  and  inverted  image  which 
it  forms  of  distant  objects  is,  in  the  Herschelian  telescope,  viewed 
directly  through  an  eye-piece,  the  back  of  the  observer  being  towards 
the  object,  and  his  face  towards  the  speculum.  This  construction  is 
only  applicable  to  very  large  specula;  as  in  instruments  of  ordinary 


MAGNIFYING  POWER   OF  TELESCOPE.  1049 

size  the  interposition  of  the  observer's  head  would  occasion  too 
serious  a  loss  of  light. 

An  arrangement  more  frequently  adopted  is  that  devised  by  Sir 
Isaac  Newton,  and 
employed  by  him  in 
the  first  reflecting  tele- 
scope ever  construct- 
ed. It  is  represented  in 
Fig.  750.  The  specu- 
lum is  at  the  bottom  of 
a  tube  whose  open  end 
is  directed  towards 
the  distant  object  to 
be  examined.  The  rays  Fig.  750.  _Newtonian  Telescope. 

1  and  2  from  one  extre- 
mity of  the  object  are  reflected  towards  a,  and  the  rays  3, 4  from  the 
other  extremity  are  reflected  towards  b.  A  real  inverted  image  a  b 
would  thus  be  formed  at  the  principal  focus  of  the  concave  speculum; 
but  a  small  plane  mirror  M  is  interposed  obliquely,  and  causes  the 
real  image  to  be  formed  at  a  b'  in  a  symmetrical  position  with 
respect  to  the  mirror  M.  The  eye-lens  O'  transforms  this  into  the 
enlarged  and  virtual  image  A'  B'. 

Magnifying  Power. — The  approximate  formula  for  the  magnifying 
power  is  the  same  as  in  the  case  of  the  refracting  telescopes  already 
described.  In  fact  the  first  image  a  b  subtends,  at  the  optical  centre 
O  (not  shown  in  the  figure)  of  the  large  speculum,  an  angle  aOb 
equal  to  the  visual  angle  for  the  naked  eye;  and  the  second  image 
a  b'  (which  is  equal  to  the  former)  subtends,  at  the  centre  of  the 
eye-piece,  an  angle  a  O'  6'  equal  to  the  angle  under  which  the  image 
is  seen  in  the  telescope.  The  magnifying  power  is  therefore  ^jy,  or, 
what  is  the  same  thing,  is  the  ratio  of  the  distance  of  a  b  from  O  to 
the  distance  of  a'  b'  from  0',  or  the  ratio  of  the  focal  length  of  the 
speculum  to  that  of  the  eye-piece. 

In  the  Gregorian  telescope,  which  was  invented  before  that  of 
Newton,  but  not  manufactured  till  a  later  date,  there  are  two  con- 
cave specula.  The  large  one,  which  receives  the  direct  rays  from 
the  object,  forms  a  real  and  inverted  image.  The  smaller  speculum 
which  is  suspended  in  the  centre  of  the  tube,  with  its  back  to  the 
object,  receives  the  rays  reflected  from  the  first  speculum,  and  forms 
a  second  real  image,  which  is  the  enlarged  and  inverted  image  of  the 


1050  VISION   AND   OPTICAL   INSTRUMENTS. 

first,  and  is  therefore  erect  as  compared  with  the  object.  This  real 
and  erect  image  is  then  magnified  by  means  of  an  eye-piece,  as  in 
the  instruments  previously  described,  the  eye-piece  being  contained 
in  a  tube  which  slides  in  a  hole  pierced  in  the  middle  of  the  large 
speculum. 

As  this  arrangement  gives  an  erect  image,  and  enables  the  observer 
to  look  directly  towards  the  object,  it  is  specially  convenient  for 
terrestrial  observation.  It  is  the  construction  almost  universally 
adopted  in  reflecting  telescopes  of  small  size. 

The  Cassegranian  telescope  resembles  the  Gregorian,  except  that 
the  second  speculum  is  convex,  and  the  image  which  the  observer 
sees  is  inverted. 

1035.  Silvered   Specula. — Achromatic  refracting   telescopes  give 
much  better  results,  both  as  regards  light  and  definition,  than  reflectors 
of  the  same  size  or  weight;  but  it  has  been  found  practicable  to 
make  specula  of  much  larger  size  than  object-glasses.  "  The  aperture 
of  Lord  Rosse's  largest  telescope  is  6  feet,  whereas  the  aperture  of 
the  largest  achromatic  telescopes  yet  constructed  is  ks^  than  two 
feet,  and  increase  of  size  involves  increased  thickness  of  glass,  and 
consequent  absorption  of  light. 

The  massiveness  which  is  found  necessary  in  the  speculum  in  order 
to  prevent  flexure,  is  a  serious  inconvenience,  as  is  also  the  necessity 
for  frequent  repolishing — an  operation  of  great  delicacy,  as  the 
slightest  change  in  the  form  of  the  surface  impairs  the  definition  of 
the  images.  Both  these  defects  have  been  to  a  certain  extent  reme- 
died by  the  introduction  of  glass  specula,  covered  in  front  with  a 
thin  coating  of  silver.  Glass  is  much  more  easily  worked  than 
speculum-metal  (which  is  remarkable  for  its  brittleness  in  casting), 
and  has  only  one-tliird  of  its  specific  gravity.  Silver  is  also  much 
superior  to  speculum-metal  in  reflecting  power;  and  as  often  as  it 
becomes  tarnished  it  can  be  removed  and  renewed,  without  liability 
to  change  of  form  in  the  reflecting  surface.1 

1036.  Measure  of  Brightness. — The  apparent  brightness  of  a  surface 
is  most  naturally  measured  by  the  amount  of  light  per  unit  area  of 
its  image  on  the  retina;  and  therefore  varies  directly  as  the  amount 
of  light  which  the  surface  sends  to  the  pupil,  and  inversely  as  the 
apparent  area  of  the  surface. 

To  avoid  complications  arising  from  the  varying  condition  of  the 

1  The  merits  of  silvered  specula  are  fully  set  forth  in  a  brochure  published  by  Mr. 
Browning,  the  optician,  entitled  A  Plea  for  Reflector!. 


BRIGHTNESS.  1051 

observer,  we  shall  leave  dilatation  and  contracfion  of  the  pupil  out 
of  account.  ' 

When  a  body  is  looked  at  through  a  small  pinhole  in  a  card  held 
close  to  the  eye,  it  appears  much  darker  than  when  viewed  in  the 
ordinary  way;  and  in  like  manner  images  formed  by  optical  instru- 
ments often  furnish  beams  of  light  too  narrow  to  fill  the  pupil.  In 
all  such  cases  it  becomes  necessary  to  distinguish  between  effective 
brightness  and  intrinsic  brightness,  the  former  being  less  than  the 
latter  in  the  same  ratio  in  which  the  cross  section  of  the  beam  which 
enters  the  pupil  is  less  than  the  area  of  the  pupil.  We  may  correctly 
speak  of  the  intrinsic  brightness  of  a  surface  for  a  particular  point 
of  the  pupil;  and  the  effective  brightness  will  in  every  case  be  the 
average  value  of  the  intrinsic  brightness  taken  over  the  whole 
pupil. 

In  the  case  of  natural  bodies  viewed  in  the  ordinary  way,  the 
distinction  may  be  neglected,  since  they  usually  send  light  in  sensibly 
equal  amounts  to  all  parts  of  the  pupil. 

To  obtain  a  numerical  measure  of  intrinsic  brightness,  let  us  denote 
by  s  a  small  area  on  a  surface  directly  facing  towards  the  eye,  or 
the  foreshortened  projection  of  a  small  area  oblique  to  the  line  of 
vision,  and  by  w  the  solid  angle  which  the  pupil  of  the  eye  subtends 
at  any  point  of  8.  Then  the  quantity  of  light  q  which  s  sends  to 
the  pupil  per  unit  time,  varies,  jointly  as  the  solid  angle  w,  the  area  s, 
and  the  intrinsic  brightness  of  s,  which  we  will  denote  by  I.  We 
may  therefore  write 


The  intrinsic  brightness  of  a  small  area  s  is  therefore  measured  by 
^,  where  q  denotes  the  quantity  of  light  which  s  emits  per  unit 

time  in  directions  limited  by  the  small  angle  of  divergence  <•». 

1037.  Applications.  —  One  of  the  most  obvious  consequences  is  that 
surfaces  appear  equally  bright  at  all  distances  in  the  same  direction, 
provided  that  no  light  is  stopped  by  the  air  or  other  intervening 
medium;  for  q  and  w  both  vary  inversely  as  the  square  of  the  dis- 
tance. The  area  of  the  image  formed  on  the  retina  in  fact  varies 
directly  as  the  amount  of  light  by  which  it  is  formed. 

Images  formed  by  Lenses,  —  Let  AB  (Fig.  751)  be  an  object,  and 
a  6  its  real  image  formed  by  the  lens  C  D,  whose  centre  is  O.  Let 
S  denote  a  small  area  at  A,  and  Q  the  quantity  of  light  which  it 
sends  to  the  lens;  also  let  s  denote  the  corresponding  area  of  the 


1052  VISION  AND   OPTICAL   INSTRUMENTS. 

image,  and  q  the  quantity  of  light  which  traverses  it.  Then  q  would 
be  identical  with  Q  if  no  light  were  stopped  by  the  lens;  the  areas,  S,  s, 
are  directly  as  the  squares  of  the  conjugate  focal  distances  O  A,  0  a; 

and  the  solid  angles  of  divergence 

f4^^-----^^^ 7l\^  ^  anc^  w'  ^or  Q  anc^  ?'  being  the 

k :^=^^— -TOJ — ^ —     <*  solid  angles  subtended  by  the  lens 

if""  ^^\,    a*  A  and  a  (for  the  plane  angle 

Kg.  75i.-Brightnes8  of  image.  cadin  the  figure  is  equal  to  the 

vertical  angle  C  a  D),  are  inversely 

as  the  squares  of  the  conjugate  focal  distances.    We  have  accordingly 

Sil  =  Sfc>.  The  intrinsic  brightness  ~  of  the  image  is  therefore 
equal  to  the  intrinsic  brightness  ~  of  the  object  except  in  so  far  as 

light  is  stopped  by  the  lens.  Precisely  similar  reasoning  applies  to 
virtual  images  formed  by  lenses.1 

In  the  case  of  images  formed  by  mirrors,  O  and  w  are  the  solid 
angles  subtended  by  the  mirror  at  the  conjugate  foci,  and  are 
inversely  as  the  squares  of  the  distances  from  the  mirror;  while  S 
and  s  are  directly  as  the  squares  of  the  distances  from  the  centre  of 
curvature;  but  these  four  distances  are  proportional  (§  967),  so  that 
the  same  reasoning  is  still  applicable.  If  the  mirror  only  reflects 
half  the  incident  light,  the  image  will  have  only  half  the  intrinsic 
brightness  of  the  object. 

If  the  pupil  is  filled  with  light  from  the  image,  the  effective 
brightness  will  be  the  same  as  the  intrinsic  brightness  thus  computed. 
If  this  condition  is  not  fulfilled,  the  former  will  be  less  than  the 
latter.  When  the  image  is  greatly  magnified  as  compared  with  the 
object,  the  angle  of  divergence  is  greatly  diminished  in  comparison 
with  the  angle  which  the  lens  or  mirror  subtends  at  the  object,  and 
often  becomes  so  small  that  only  a  small  part  of  the  pupil  is  utilized. 
This  is  the  explanation  of  the  great  falling  off  of  light  which  is 
observed  in  the  use  of  high  magnifying  powers,  both  in  microscopes 
and  telescopes. 

1  For  refraction  out  of  a  medium  of  index  Hi  into  another  of  index  ^  we  have  by 

§  1015,  equation  (13),  /tt :  ^  :  :  — — *  :  _ ---*.     But  since  «„  st  are  the  areas  of  correspond- 
O  x  i      O  Ira 

ing  parts  of  object  and  image,  we  have  st  :  s2 : :  C  PIS  :  C  P22,  and  since  a>i,  u.2  are  the  solid 
angles  subtended  at  P1?  Ps  by  one  and  the  same  portion  of  the  bounding  surface,  we  have 

o>! :  w3 : :  AP2*  :  A  Pj8.     Therefore  -%—  '•  -*-  :  :  MI*  :  fa3.     The  intrinsic  brightnesses  of  a 

succession  of  images  in  different  media  are  therefore  directly  as  the  squares  of  the  absolute 
indices.  On  this  point  see  Phil.  Mag.,  March,  1888,  p.  216, 


BRIGHTNESS.  1053 

1038.  Brightness  of  Image  in  a  Telescope. — It  has  been  already 
pointed  out  (§  1031)  that  in  most  forms  of  telescope  (the  Galilean 
being  an  exception),  there  is  a  certain  position,  a  little  behind  the 
eye-piece,  at  which  a  well-defined  bright  spot  is  formed  upon  a 
screen  held  there  while  the  telescope  is  directed  to  any  distant  source 
of  light.  It  has  also  been  pointed  out  that  this  spot  is  the  image, 
formed  by  the  eye-piece,  of  the  opening  which  is  filled  by  the  object- 
glass,  and  that  the  magnifying  power  of  the  instrument  is  the  ratio 
of  the  size  of  the  object-glass  to  the  size  of  this  bright  spot. 

Let  s  denote  the  diameter  of  the  bright  spot,  o  the  diameter  of  the 

object-glass,  e  the  diameter  of  the  pupil  of  the  eye;  then  -?-  is  the 

linear  magnifying  power. 

We  shall  first  consider  the  case  in  which  the  spot  exactly  covers 
the  pupil  of  the  observer's  eye,  so  that  s=e.  Then  the  whole 
light  which  traverses  the  telescope  from  a  distant  object  enters  the 
eye;  and  if  we  neglect  the  light  stopped  in  the  telescope,  this  is 

the  whole  light  sent  by  the  object  to  the  object-glass,  and  is  (-M 
times  that  which  would  be  received  by  the  naked  eye.  The  magni- 
fication of  apparent  area  is  (  —  J ,  which,  from  the  equality  of  8  and  e, 

is  the  same  as  the  increase  of  total  light.  The  brightness  is  therefore 
the  same  as  to  the  naked  eye. 

Next,  let  s  be  greater  than  e,  and  let  the  pupil  occupy  the  central 
part  of  the  spot.  Then,  since  the  spot  is  the  image  of  the  object- 
glass,  we  may  divide  the  object-glass  into  two  parts — a  central  part 
whose  image  coincides  with  the  pupil,  and  a  circumferential  part 
whose  image  surrounds  the  pupil.  All  rays  from  the  object  which 
traverse  the  central  part,  traverse  its  image,  and  therefore  enter  the 
pupil;  whereas  rays  traversing  the  circumferential  part  of  the  object- 
glass,  traverse  the  circumferential  part  of  the  image,  and  so  are 
wasted.  The  area  of  the  central  part  (whether  of  the  object-glass 
or  of  its  image)  is  to  the  whole  area  as  e"' :  s2;  and  the  light  which 

the  object  sends  to  the  central  portion,  instead  of  being  (  -^  J  times 
that  which  would  be  received  by  the  naked  eye,  is  only  (yj  times. 

But  (y  j  is  the  magnification  of  apparent  area.  Hence  the  bright- 
ness is  the  same  as  to  the  naked  eye.  In  these  two  cases,  effective 
and  intrinsic  brightness  are  the  same. 


1054  VISION   AND   OPTICAL   INSTRUMENTS. 

\ 

Lastly  (and  this  is  by  far  the  most  common  case  in  practice),  let 
s  be  less  than  e.  Then  no  light  is  wasted,  but  the  pupil  is  not  filled. 

The  light  received  is  (y)2  times  that  which  the  naked  eye  would 

receive;  and  the  magnification  of  apparent  area  is  (7)  .  The  effec- 
tive brightness  of  the  image,  is  to  the  brightness  of  the  object  to  the 
naked  eye,  as  (  ^-)2  :  (  j)2;  that  is,  as  s'2 : e2;  that  is,  as  the  area  of  the 

bright  spot  to  the  whole  area  of  the  pupil. 

To  correct  for  the  light  stopped  by  reflection  and  imperfect  trans- 
parency, we  have  simply  to  multiply  the  result  in  each  case  by  a 
proper  fraction,  expressing  the  ratio  of  the  transmitted  to  the  incident 
light.  This  ratio,  for  the  central  parts  of  the  field  of  view,  is  about 
O85  in  the  best  achromatic  telescopes.  In  such  telescopes,  therefore, 
the  brightness  of  the  image  cannot  exceed  0'85  of  the  brightness  of 
the  object  to  the  naked  eye.  It  will  have  this  precise  value,  when 
the  magnifying  power  is  equal  to  or  less  than  ^;  and  from  this 
point  upwards  will  vary  inversely  as  the  square  of  the  linear  mag- 
nification. 

The  same  formulae  apply  to  reflecting  telescopes,  o  denoting  now 
the  diameter  of  the  large  speculum  which  serves  as  objective;  but 
the  constant  factor  is  usually  considerably  less  than  0'85. 

It  may  be  accepted  as  a  general  principle  in  optics,  that  while  it 
is  possible,  by  bad  focussing  or  instrumental  imperfections,  to  obtain 
a  confused  image  whose  brightness  shall  be  intermediate  between 
the  brightest  and  the  darkest  parts  of  the  object,  it  is  impossible, 
by  any  optical  arrangement  whatever,  to  obtain  an  image  whose 
brightest  part  shall  surpass  the  brightest  part  of  the  object. 

1039.  Brightness  of  Stars. — There  is  one  important  case  in  which 
the  foregoing  rules  regarding  the  brightness  of  images  become 
nugatory.  The  fixed  stars  are  bodies  which  subtend  at  the  earth 
angles  smaller  than  the  minimum  visibile,  but  which,  on  account 
of  their  excessive  brightness,  appear  to  have  a  sensible  angular 
diameter.  This  is  an  instance  of  irradiation,  a  phenomenon  mani- 
fested by  all  bodies  of  excessive  brightness,  and  consisting  in  an 
extension  of  their  apparent  beyond  their  actual  boundary.  What 
is  called,  in  popular  language,  a  bright  star,  is  a  star  which  sends  a 
large  total  amount  of  light  to  the  eye. 

Denoting  by  a  the  ratio  of  the  transmitted  to  the  whole  incident 
light,  a  ratio  which,  as  we  have  seen,  is  about  0'85  in  the  most 


BRIGHTNESS.  1055 

favourable  cases,  and  calling  the  light  which  a  star  sends  to  the 
naked  eye  unity,  the  light  perceived  in  its  image  will  be  a  (-J-)  ,  or 

a  x  square  of  linear  magnification,  if  the  bright  spot  is  as  large  as 
the  pupil.  When  the  eye-piece  is  changed,  increase  of  power  dimin- 
ishes the  size  of  the  spot,  and  increases  the  light  received  by  the  eye, 
until  the  spot  is  reduced  to  the  size  of  the  pupil.  After  this,  any 
further  magnification  has  no  effect  on  the  quantity  of  light  received, 

its  constant  value  being  a  ( -M  • 

The  value  of  this  last  expression,  or  rather  the  value  of  a  o2,  is  the 
measure  of  what  is  called  the  space-penetrating  poiuer  of  a  telescope; 
that  is  to  say,  the  power  of  rendering  very  faint  stars  visible;  and 
it  is  in  this  respect  that  telescopes  of  very  large  aperture,  notably 
the  great  reflector  of  Lord  Rosse,  are  able  to  display  their  great 
superiority  over  instruments  of  moderate  dimensions. 

We  have  seen  that  the  total  light  in  the  visible  image  of  a  star 
remains  unaltered,  by  increase  of  power  in  the  eye-piece  above  a 
certain  limit.  But  the  visibility  of  faint  stars  in  a  telescope  is  pro- 
moted by  darkening  the  back-ground  of  sky  on  which  they  are  seen. 
Now  the  brightness  of  this  back-ground  varies  directly  as  s2,  or 
inversely  as  the  square  of  the  linear  magnification  (s  being  supposed 
less  than  e).  Hence  it  is  advantageous,  in  examining  very  faint 
stars,  to  employ  eye-pieces  of  sufficient  power  to  render  the  bright 
spot  much  smaller  than  the  pupil  of  the  eye. 

1040.  Images  on  a  Screen. — Thus  far  we  have  been  speaking  of  the 
brightness  of  images  as  viewed  directly.  Images  cast  upon  a  screen 
are,  as  a  matter  of  fact,  much  less  brilliant;  partly  because  the  screen 
sends  out  light  in  all  directions,  and  therefore  through  a  much  larger 
solid  angle  than  that  formed  by  the  beam  incident  on  the  screen, 
and  partly  because  some  of  the  incident  light  is  absorbed. 

Let  A  be  the  area  of  the  object,  which  we  suppose  to  face  directly 
towards  the  lens  by  which  the  image  is  thrown  upon  the  screen,  a 
the  area  of  the  image,  and  D,  d  their  respective  distances  from  the 
lens.  Then  if  I  denote  the  intrinsic  brightness  of  the  object,  the 
light  sent  from  A  to  the  lens  will  be  the  product  of  I A  by  the 
solid  angle  which  the  lens  subtends  at  the  object.  This  solid  angle 

will  be  —7,  if  L  denote  the  area  of  the  lens.     I A  -^  is  therefore  the 

light  sent  by  the  object  to  the  lens,  and  if  we  neglect  reflection  and 
absorption  all  this  light  falls  upon  the  image.  The  light  which  falls 


1056  VISION   AND   OPTICAL   INSTRUMENTS. 

071  unit  area  of  the  image  is  therefore  I  —  -j^,  that  is  I  -^ ;  it  is 
therefore  the  same  as  if  the  lens  were  a  source  of  light  of  brightness 
I.  Accordingly,  if  the  image  of  a  lamp  flame  be  thrown  upon  the 
pupil  of  an  observer's  eye,  and  be  large  enough  to  cover  the  pupil, 
he  will  see  the  lens  filled  with  light  of  a  brightness  equal  to  that  of 
the  flame  seen  directly. 

1041.  Field  of  View  in  Astronomical  Telescope. — Let  pmnq  (Fig. 
752)  be  the  common  focal  plane  of  the  object-glass  and  eye-glass. 

Draw  B  a,  A  b  joining  the  high- 
est points  of  both,  and  the  lowest 
points  of  both;  also  A  a,  B  b  join- 
ing the  highest  point  of  each  with 
the  lowest  point  of  the  other. 

Fig.  752.-FieldofView.  r 

Evidently  Ba,  A  6  will   be  the 

boundaries  of  the  beam  of  light  transmitted  through  the  telescope, 
and  therefore  the  points  p  and  q  in  which  these  lines  intersect  the 
focal  plane,  will  be  the  extremities  of  that  part  of  the  real  image 
which  sends  rays  to  the  eye.  The  angle  subtended  by  p  q  at  the 
centre  of  the  object-glass  will  therefore  be  the  angular  diameter  of 
the  complete  field  of  view.  But  the  outer  portions  of  this  field  will 
be  less  bright  than  the  centre,  and  the  full  amount  of  brightness,  as 
calculated  in  §  1038  for  the  case  in  which  the  "  bright  spot "  is 
smaller  than  the  pupil,  will  belong  only  to  the  portion  m  n  bounded 
by  the  cross-lines  A  a,  B  6;  for  all  the  rays  sent  by  the  object-glass 
through  the  part  mn  traverse  the  eye-glass,  and  therefore  the 
bright  spot,  whereas  some  of  the  rays  sent  by  the  object-glass  to 
any  point  between  m  and  p,  or  between  n  and  q  pass  wide  of  the 
eye-glass  and  therefore  do  not  reach  the  bright  spot.  The  complete 
field  of  view,  as  seen  by  an  eye  whose  pupil  includes  the  bright  spot, 
accordingly  consists  of  a  central  disc  mn  of  full  brightness,  sur- 
rounded by  a  ring  extending  to  p  and  q  whose  brightness  gradually 

diminishes  from  full  brightness 
at  its  junction  with  the  disc  to 
zero  at  its  outer  boundary.  This 
ring  is  called  the  "  ragged  edge," 
and  is  put  out  of  sight  in  actual 

Fig.  753.— Calculation  of  Field. 

telescopes  by  an  opaque  stop  of 

annular  form  in  the  focal  plane.  The  angular  diameter  of  the  field 
of  view,  excluding  the  ragged  edge,  will  be  equal  to  the  angle 
which  mn  subtends  at  the  centre  of  the  object-glass. 


FIELD   OF   VIEW.  1057 

To  calculate  the  length  of  m  n,  join  D,  d,  the  centres  of  the 
object-glass  and  eye-glass  (Fig.  753).  The  joining  line  will  obviously 
pass  through  the  intersection  of  A  a,  B6,  and  also  through  the 
middle  point  of  mn.  Draw  a  parallel  to  this  line  through  m. 
Then,  by  comparing  the  similar  triangles  of  which  a  m,  A  m  are  the 
hypotenuses,  we  have 

ad-  mo    :   od    ::    AD  +  mo    :    Do. 

Hence,  multiplying  extremes  and  means,  and  denoting  the  focal 
lengths  D  o,  o  d  by  F,  /,  we  have 

F  (ad-mo)  =f  (AD  +  mo), 

whence 


This  is  the  radius  of  the  real  image,  excluding  the  ragged  edge;  and 
the  angular  radius  of  the  field  of  view  will  be 

mo  _  F.ad-f.AV 
F    =         F 
ad 


The  first  term  ^—  -,  is  the  angle  which  the  radius  of  the  eye-glass 

subtends  at  the  object-glass.  But,  it  is  obvious  from  Fig.  752  that 
the  line  aD  would  bisect  Dip.  Hence  the  second  term  represents 
half  the  breadth  of  the  ragged  edge,  and  the  whole  field  of  view, 
including  the  ragged  edge,  has  an  angular  radius 

ad         /.AD 


1042.  Cross-wires  of  Telescopes.  —  We  have  described  in  §  1010  a 
mode  of  marking  the  place  of  a  real  image  by  means  of  a  cross  of 
threads.  When  telescopes  are  employed  to  assist  in  the  measure- 
ment of  angles,  a  contrivance  of  this  kind  is  almost  always  intro- 
duced. A  cross  of  silkworm  threads,  in  instruments  of  low  power, 
or  of  spider  threads  in  instruments  of  higher  power,  is  stretched 
across  a  metallic  frame  just  in  front  of  the  eye-piece.  The  observer 
must  first  adjust  the  eye-piece  for  distinct  vision  of  this  cross,  and 
must  then  (in  the  case  of  theodolites  and  other  surveying  instru- 
ments) adjust  the  distance  of  the  object-glass  until  the  object  which 
is  to  be  observed  is  also  seen  distinctly  in  the  telescope.  The  image 
of  the  object  will  then  be  very  nearly  in  the  plane  of  the  cross.  If 
it  is  not  exactly  in  the  plane,  parallactic  displacement  will  be 
observed  when  the  eye  is  shifted,  and  this  must  be  cured  by  slightly 
67 


1058  VISION   AND   OPTICAL  INSTRUMENTS. 

altering  the  distance  of  the  object-glass.  When  the  adjustment  has 
been  completed,  the  cross  always  marks  one  definite  point  of  the 
object,  however  the  eye  be  shifted.  This  coincidence  will  not  be 
disturbed  by  pushing  in  or  pulling  out  the  eye-piece;  for  the  frame 
which  carries  the  cross  is  attached  to  the  body  of  the  telescope,  and 
the  coincidence  of  the  cross  with  a  point  of  the  image  is  real,  so  that 
it  could  be  observed  by  the  naked  eye,  if  the  eye-piece  were  removed. 
The  adjustment  of  the  eye-piece  merely  serves  to  give  distinct  vision, 
and  this  will  be  obtained  simultaneously  for  both  the  cross  and  the 
object. 

1043.  Line  of  Collimation. — The  employment  of  cross-wires  (as 
these  crossing  threads  are  called)  enormously  increases  our  power  of 
making  accurate  observations  of  direction,  and  constitutes  one  of  the 
greatest  advantages  of  modern  over  ancient  instruments. 

The  line  which  is  regarded  as  the  line  of  sight,  or  as  the  direction 
in  which  the  telescope  is  pointed,  is  called  the  line  of  collimation. 
If  we  neglect  the  curvature  of  rays  due  to  atmospheric  refraction, 
we  may  define  it  as  the  line  joining  the  cross  to  the  object  ivhese 
image  falls  on  it.  More  rigorously,  the  line  of  collimation  is  the 
line  joining  the  cross  to  the  optical  centre  of  the  object-glass.  When 
it  is  desired  to  adjust  the  line  of  collimation, — for  example,  to  make 
it  truly  perpendicular  to  the  horizontal  axis  on  which  the  telescope 
is  mounted,  the  adjustment  is  performed  by  shifting  the  frame  which 
carries  the  wires,  slow-motion  screws  being  provided  for  this  purpose. 
Telescopes  for  astronomical  observation  are  often  furnished  with  a 
number  of  parallel  wires,  crossed  by  one  or  two  in  the  transverse 
direction ;  and  the  line  of  collimation  is  then  defined  by  reference  to 
an  imaginary  cross,  which  is  the  centre  of  mean  position  of  all  the 
actual  crosses. 

1044.  Micrometers. — Astronomical   micrometers   are   of   various 
kinds,  some  of  them  serving  for  measuring  the  angular  distance  be- 
tween two  points  in  the  same  field  of  view,  and  others  for  measur- 
ing their  apparent  direction  from  one  another.    They  often  consist  of 
spider  threads  placed  in  the  principal  focus  of  the  object-glass,  so  as  to 
be  in  the  same  plane  as  the  images  of  celestial  objects,  one  or  more  of 
the  threads  being  movable  by  means  of  slow-motion  screws,  furnished 
with  graduated  circles,  on  which  parts  of  a  turn  can  be  read  off. 

One  of  the  commonest  kinds  consists  of  two  parallel  threads,  which 
can  thus  be  moved  to  any  distance  apart,  and  can  also  be  turned 
round  in  their  own  plane. 


CHAPTER    LXXII. 


DISPERSION.      STUDY   OF  SPECTRA. 


1045.  Newtonian  Experiment. — In  the  chapter  on  refraction,  we 
have  postponed  the  discussion  of  one  important  phenomenon  by 
which  it  is  usually  accompanied,  and  which  we  must  now  proceed  to 
explain.  The  following  experiment,  which  is  due  to  Sir  Isaac  Newton, 
will  furnish  a  fitting  introduction  to  the  subject. 

On  an  extensive  background  of  black,  let  three  bright  strips  be 
laid  in  line,  as  in  the  left-hand  part  of  Fig.  754,  and  looked  at 
through  a  prism  with  its  refracting  edge  parallel  to  the  strips.  "We 


Fig.  754.— Spectra  of  White  and  Coloured  Strips. 

shall  suppose  the  edge  to  be  upward,  so  that  the  image  is  raised  above 
the  object.  The  images,  as  represented  in  the  right-hand  part  of 
Fig.  754,  will  have  the  same  horizontal  dimensions  as  the  strips,  but 
will  be  greatly  extended  in  the  vertical  direction;  and  each  image, 
instead  of  having  the  uniform  colour  of  the  strips  from  which  it  is 
derived,  will  be  tinted  with  a  gradual  succession  of  colours  from  top 
to  bottom.  Such  images  are  called  spectra. 

If  one  of  the  strips  (the  middle  one  in  the  figure)  be  white,  its 
spectrum  will  contain  the  following  series  of  colours,  beginning  at 
the  top :  violet,  blue,  green,  yellow,  orange,  red. 


1060  DISPERSION.      STUDY   OF  SPECTRA. 

If  one  of  the  strips  be  blue  (the  left-hand  one  in  the  figure),  its 
image  will  present  bright  colours  at  the  upper  end;  and  these  will 
be  identical  with  the  colours  adjacent  to  them  in  the  spectrum  of 
white.  The  colours  which  form  the  lower  part  of  the  spectrum  of 
white  will  either  be  very  dim  and  dark  in  the  spectrum  of  blue,  or 
will  be  wanting  altogether,  being  replaced  by  black. 

If  the  other  strip  be  red,  its  image  will  contain  bright  colours  at 
the  lower  or  red  end,  and  those  which  belong  to  the  upper  end  of  the 
spectrum  of  white  will  be  dim  or  absent.  Every  colour  that  occurs 
in  the  spectrum  of  blue  or  of  red  will  also  be  found,  and  in  the  same 
horizontal  line,  in  the  spectrum  of  white. 

If  we  employ  other  colours  instead  of  blue  or  red,  we  shall  obtain 
analogous  results;  every  colour  will  be  found  to  give  a  spectrum 
which  is  identical  with  part  of  the  spectrum  of  white,  both  as  regards 
colour  and  position,  but  not  generally  as  regards  brightness. 

We  may  occasionally  meet  with  a  body  whose  spectrum  consists 
only  of  one  colour.  The  petals  of  some  kinds  of  convolvulus  give  a 
spectrum  consisting  only  of  blue,  and  the  petals  of  nasturtium  give 
only  red. 

1046.  Composite  Nature  of  Ordinary  Colours.  —  This  experiment 
shows  that  the  colours  presented  by  the  great  majority  of  natural 
bodies  are  composite.  When  a  colour  is  looked  at  with  the  naked 
eye,  the  sensation  experienced  is  the  joint  effect  of  the  various  elemen- 
tary colours  which  compose  it.  The  prism  serves  to  resolve  the  colour 
into  its  components,  and  exhibit  them  separately.  The  experiment 
also  shows  that  a  mixture  of  all  the  elementary  colours  in  proper 
proportions  produces  white. 

.  1047.  Solar  Spectrum. — The  coloured  strips  in  the  foregoing  experi- 
ment may  be  illuminated  either  by  daylight  or  by  any  of  the  ordinary 
sources  of  artificial  light.  The  former  is  the  best,  as  gas-light  and 
candle-light  are  very  deficient  in  blue  and  violet  rays. 

Colour,  regarded  as  a  property  of  a  coloured  (opaque)  body,  is  the 
power  of  selecting  certain  rays  and  reflecting  them  either  exclusively 
or  in  larger  proportion  than  others.  The  spectrum  presented  by  a 
body  viewed  by  reflected  light,  as  ordinary  bodies  are,  can  thus  only 
consist  of  the  rays,  or  a  selection  of  the  rays,  by  which  the  body  is 
illuminated. 

A  beam  of  solar  light  can  be  directly  resolved  into  its  constituents 
by  the  following  experiment,  which  is  also  due  to  Newton,  and  was 
the  first  demonstration  of  the  composite  character  of  solar  light. 


SOLAR   SPECTRUM. 


1061 


Let  a  beam  of  sun-light  be  admitted  through  a  small  opening  into 
a  dark  room.  If  allowed  to  fall  normally  on  a  white  screen,  it  pro- 
duces (§  938)  a  round  white  spot,  which  is  an  image  of  the  sun.  Now 
let  a  prism  be  placed  in  its  path  edge-downwards,  as  in  Fig.  755;  the 


Fig.  755.  —Solar  Spectrum  by  Newton's  Method. 

beam  will  thus  be  deflected  upwards,  and  at  the  same  time  resolved 
into  its  component  colours.  The  image  depicted  on  the  screen  will 
be  a  many-coloured  band,  resembling  the  spectrum  of  white  described 
in  §  1045.  -It  will  be  of  uniform  width,  and  rounded  off  at  the  ends, 
being  in  fact  built  up  of  a  number  of  overlapping  discs,  one  for  each 
kind  of  elementary  ray.  It  is  called  the  solar  spectrum,. 

The  rays  which  have  undergone  the  greatest  deviation  are  the 
violet.  They  occupy  the  upper  end  of  the  spectrum  in  the  figure. 
Those  which  have  undergone  the  least  deviation  are  the  red.  Of  all 
visible  rays,  the  violet  are  the  most,  and  the  red  the  least  refrangible; 
and  the  analysis  of  light  into  its  components  by  means  of  the  prism  is 
due  to  difference  of  refrangibility.  If  a  small  opening  is  made  in  the 
screen,  so  as  to  allow  rays  of  only  one  colour  to  pass,  it  will  be  found, 


1062  DISPERSION.      STUDY   OF  SPECTRA. 

on  transmitting  these  through  a  second  prism  behind  the  screen,  as 
in  Fig.  755,  that  no  further  analysis  can  be  effected,  and  the  whole 
of  the  image  formed  by  receiving  this  transmitted  light  on  a  second 
screen  will  be  of  this  one  colour. 

1048.  Mode  of  obtaining  a  Pure  Spectrum. — The  spectra  obtained 
by  the  methods  above  described  are  built  up  of  a  number  of  over- 
lapping images  of  different  colours.  To  prevent  this  overlapping,  and 
obtain  each  elementary  colour  pure  from  all  admixture  with  the  rest, 
we  must  in  the  first  place  employ  as  the  object  for  yielding  the 
images  a  very  narrow  line;  and  in  the  second  place  we  must  take 
care  that  the  images  which  we  obtain  of  this  line  are  not  blurred, 
but  have  the  greatest  possible  sharpness.  A  spectrum  possessing 
these  characteristics  is  called  pure. 

The  simplest  mode  of  obtaining  a  pure  spectrum  consists  in 
looking  through  a  prism  at  a  fine  slit  in  the  shutter  of  a  dark  room. 
The  edges  of  the  prism  must  be  parallel  to  the  slit,  and  its  distance 
from  the  slit  should  be  five  feet  or  upwards.  The  observer,  placing 
his  eye  close  to  the  prism,  will  see  a  spectrum;  and  he  should  rotate 
the  prism  on  its  axis  until  he  has  brought  this  spectrum  to  its 
smallest  angular  distance  from  the  real  slit,  of  which  it  is  the  image. 
Let  E  (Fig.  756)  be  the  position  of  the  eye,  S  that  of  the  slit. 
Then  the  extreme  red  and  violet  images  of  the  slit  will  be  seen  at 
R,  V,  at  distances  from  the  prism  sensibly  equal  to  the  real  distance 
of  S  (§  997);  and  the  other  images,  which  compose  the  remainder 

of  the  spectrum,  will  occupy  posi- 
tions between  R  and  V.  The 
spectrum,  in  this  mode  of  operat- 
ing, is  virtual. 

To  obtain  a  real  spectrum  in  a 
state  of  purity,  a  convex  lens 
must  be  employed.  Let  the  lens 
L  (Fig.  757)  be  first  placed  in 
such  a  position  as  to  throw  a 
sharp  image  of  the  slit  S  upon 

Fig.  75G.-Arrangement  for  seeing  a  Pure  SpecTrmn.   ft  SCreen  at  L      Next   let  a  Prism 

P  be  introduced  between  the  lens 

and  screen,  and  rotated  on  its  axis  till  the  position  of  minimum  devia- 
tion is  obtained,  as  shown  by  the  movements  of  the  impure  spectrum 
which  travels  about  the  walls  of  the  room.  Then  if  the  screen  be 
moved  into  the  position  R  V,  its  distance  from  the  prism  being  the 


PURE  SPECTRUM.  1063 

same  as  before,  a  pure  spectrum  will  be  depicted  upon  it.  A  similar 
result  can  be  obtained  by  placing  the  prism  between  the  lens  and 
the  slit,  but  the  adjustments  are  rather  more  troublesome.  Direct 


Fig.  757. — Arrangement  for  Pure  Spectrum  on  Screen. 


sun-light,  or  sun-light  reflected  from  a  mirror  placed  outside  the 
shutter,  is  necessary  for  this  experiment,  as  sky-light  is  not  suffi- 
ciently powerful.  It  is  usual,  in  experiments  of  this  kind,  to  em- 
ploy a  movable  mirror  called  a  heliostat,  by  means  of  which  the  light 
can  be  reflected  in  any  required  direction.  Sometimes  the  move- 
ments of  the  mirror  are  obtained  by  hand;  sometimes  by  an  ingenious 
clock-work  arrangement,  which  causes  the  reflected  beam  to  keep  its 
direction  unchanged  notwithstanding  the  progress  of  the  sun  through 
the  heavens. 

The  advantage  of  placing  the  prism  in  the  position  of  minimum 
deviation  is  twofold.  First,  the  adjustments  are  facilitated  by  the 
equality  of  conjugate  focal  distances,  which  subsists  in  this  case  and 
in  this  only.  Secondly  and  chiefly,  this  is  the  only  position  in  which 
the  images  are  not  blurred.  In  any  other  position  it  can  be  shown1 
that  a  small  cone  of  homogeneous  incident  rays  is  no  longer  a  cone 
(that  is,  its  rays  do  not  accurately  pass  through  one  point)  after 
transmission  through  the  prism. 

The  method  of  observation  just  described  was  employed  by 
Wollaston,  in  the  earliest  observations  of  a  pure  spectrum  ever 
obtained.  Fraunhofer,  a  few  years  later,  independently  devised  the 
same  method,  and  carried  it  to  much  greater  perfection.  Instead  of 
looking  at  the  virtual  image  with  the  naked  eye,  he  viewed  it  through 
a  telescope,  which  greatly  magnified  it,  and  revealed  several  features 
never  before  detected.  The  prism  and  telescope  were  at  a  distance 
of  24  feet  from  the  slit. 

1  Parkinson's  Optics,  §  96.     Cor.  2. 


10G4  DISPERSION.      STUDY   OF   SPECTHA. 

1049.  Dark  Lines  in  the  Solar  Spectrum.— When  a  pure  spectrum 
of  solar  light  is  examined  by  any  of  these  methods,  it  is  seen  to  be 
traversed  by  numerous  dark  lines,  constituting,  if  we  may  so  say, 
dark  images  of  the  slit.  Each  of  these  is  an  indication  that  a  par- 
ticular kind  of  elementary  ray  is  wanting1  in  solar  light.  Every 
elementary  ray  that  is  present  gives  its  own  image  of  the  slit  in  its 
own  peculiar  colour;  and  these  images  are  arranged  in  strict  con- 
tiguity, so  as  to  form  a  continuous  band  of  light  passing  by  perfectly 
gradual  transitions  through  the  whole  range  of  simple  colour,  except 
at  the  narrow  intervals  occupied  by  the  dark  lines.  Fig.  1,  Plate  III., 
is  a  rough  representation  of  the  appearance  thus  presented.  If  the 
slit  is  illuminated  by  a  gas  flame,  or  by  any  ordinary  lamp,  instead 
of  by  solar  light,  no  such  lines  are  seen,  but  a  perfectly  continuous 
spectrum  is  obtained.  The  dark  lines  are  therefore  not  characteristic 
of  light  in  general,  but  only  of  solar  light. 

Wollaston  saw  and  described  some  of  the  more  conspicuous  of 
them.  Fraunhofer  counted  about  600,  and  marked  the  places  of  354 
upon  a  map  of  the  spectrum,  distinguishing  some  of  the  more  con- 
spicuous by  the  names  of  letters  of  the  alphabet,  as  indicated  in  fig.  1.  \ 
These  lines  are  constantly  referred  to  as  reference  marks  for  the 
accurate  specification  of  different  portions  of  the  spectrum.  They 
always  occur  in  precisely  the  same  places  as  regards  colour,  but  do 
not  retain  exactly  the  same  relative  distances  one  from  another,  when 
prisms  of  different  materials  are  employed,  different  parts  of  the 
spectrum  being  unequally  expanded  by  different  refracting  sub- 
stances.2 The  inequality,  however,  is  not  so  great  as  to  introduce 
any  difficulty  in  the  identification  of  the  lines. 

The  dark  lines  in  the  solar  spectrum  are  often  called  Fraunhofer's 
lines.  Fraunhofer  himself  called  them  the  "fixed  lines." 

1050.  Invisible  Bays  of  the  Spectrum. — The  brightness  of  the  solar 
spectrum,  however  obtained,  is  by  no  means  equal  throughout,  but 
is  greatest  between  the  dark  lines  D  and  E;  that  is  to  say,  in  the 
yellow  and  the  neighbouring  colours  orange  and  light  green;  and 
falls  off  gradually  on  both  sides. 

The  heating  effect  upon  a  small  thermometer  or  thermopile  in- 
creases in  going  from  the  violet  to  the  red,  and  still  continues  to 
increase  for  a  certain  distance  beyond  the  visible  spectrum  at  the 
red  end.  Prisms  and  lenses  of  rock-salt  should  be  employed  for  this 

1  Probably  not  absolutely  wanting,  but  so  feeble  as  to  appear  black  by  contrast. 
a  This  property  is  called  the  irrationality  of  dispersion. 


PHOSPHORESCENCE.  10G5 

investigation,  as  glass  largely  absorbs  the  invisible  rays  which  lie 
beyond  the  red. 

When  the  spectrum  is  thrown  upon  the  sensitized  paper  employed 
in  photography,  the  action  is  very  feeble  in  the  red,  strong  in  the 
blue  and  violet,  and  is  sensible  to  a  great  distance  beyond  the  violet 
end.  When  proper  precautions  are  taken  to  insure  a  very  pure 
spectrum,  the  photograph  reveals  the  existence  of  dark  lines,  like 
those  of  Fraunhofer,  in  the  invisible  ultra-violet  portion  of  the  spec- 
trum. The  strongest  of  these  have  been  named  L,  M,  N,  0,  P. 

1051.  Phosphorescence  and  Fluorescence. —  There  are  some  sub- 
stances which,  after  being  exposed  in  the  sun,  are  found  for  a  long 
time  to  appear  self-lurninous  when  viewed  in  the  dark,  and  this 


Fig.  758. — Becquerel's  Phosphoroscope. 


without  any  signs  of  combustion  or  sensible  elevation  of  temperature. 
Such  substances  are  called  phosphorescent.  Sulphuret  of  calcium  and 
sulphuret  of  barium  have  long  been  noted  for  this  property,  and  have 
hence  been  called  respectively  Cantons  phosphorus,  and  Bologna 


1066  DISPERSION.      STUDY   OF  SPECTRA. 

phosphorus.  The  phenomenon  is  chiefly  due  to  the  action  of  the 
violet  and  ultra-violet  portion  of  the  sun's  rays. 

More  recent  investigations  have  shown  that  the  same  property 
exists  in  a  much  lower  degree  in  an  immense  number  of  bodies,  their 
phosphorescence  continuing,  in  most  cases,  only  for  a  fraction  of  a 
second  after  their  withdrawal  from  the  sun's  rays.  E.  Becquerel  has 
contrived  an  instrument,  called  the  phosphoroscope,  which  is  ex- 
tremely appropriate  for  the  observation  of  this  phenomenon.  It  is 
represented  in  Fig.  758.  Its  most  characteristic  feature  is  a  pair  of 
rigidly  connected  discs  (Fig.  759),  each  pierced  with  four  openings, 
those  of  the  one  being  not  opposite  but  midway  between  those  of  the 
other.  This  pair  of  discs  can  be  set  in  very 
rapid  rotation  by  means  of  a  series  of  wheels 
and  pinions.  The  body  to  be  examined  is 
Attached  to  a  fixed  stand  between  the  two 
discs,  so  that  it  is  alternately  exposed  on 
opposite  sides  as  the  discs  rotate.  One  side 
is  turned  towards  the  sun,  and  the  other 
towards  the  observer,  who  accordingly  only 
sees  the  body  when  it  is  not  exposed  to  the 
sun's  rays.  The  cylindrical  case  within 
Discs  of  Phosjhoroscope.  which  the  discs  revolve,  is  fitted  into  a  hole 
in  the  shutter  of  a  dark  room,  and  is  pierced 

with  an  opening  on  each  side  exactly  opposite  the  position  in  which 
the  body  is  fixed.  The  body,  if  not  phosphorescent,  will  never  be  seen 
by  the  observer,  as  it  is  always  in  darkness  except  when  it  is  hidden 
by  the  intervening  disc.  If  its  phosphorescence  lasts  as  long  as  an 
eighth  part  of  the  time  of  one  rotation,  it  will  become  visible  in  the 
darkness. 

Nearly  all  bodies,  when  thus  examined,  show  traces  of  phosphor- 
escence, lasting,  however,  in  some  cases,  only  for  a  ten-thousandth 
of  a  second. 

The  phenomenon  of  fluorescence,  which  is  illustrated  in  Plate  II. 
accompanying  §  817,  appears  to  be  essentially  identical  with  phos- 
phorescence. The  former  name  is  applied  to  the  phenomenon,  if  it 
is  observed  while  the  body  is  actually  exposed  to  the  source  of  light, 
the  latter  to  the  effect  of  the  same  kind,  but  usually  less  intense, 
which  is  observed  after  the  light  from  the  source  is  cut  off.  Both 
forms  of  the  phenomenon  occur  in  a  strongly-marked  degree  in  the 
same  bodies.  Canary-glass,  which  is  coloured  with  oxide  of  uranium,  is 


FLUORESCENCE.  1067 

a  very  convenient  material  for  the  exhibition  of  fluorescence.  A  thick 
piece  of  it,  held  in  the  violet  or  ultra-violet  portion  of  the  solar 
spectrum,  is  filled  to  the  depth  of  from  ^  to  ^  of  an  inch  with  a  faint 
nebulous  light.  A  solution  of  sulphate  of  quinine  is  also  frequently 
employed  for  exhibiting  the  same  effect,  the  luminosity  in  this  case 
being  bluish.  If  the  solar  spectrum  be  thrown  upon  a  screen  freshly 
washed  with  sulp'hate  of  quinine,  the  ultra-violet  portion  will  become 
visible  by  fluorescence;  and  if  the  spectrum  be  very  pure,  the  pre- 
sence of  dark  lines  in  this  portion  will  be  detected. 

The  light  of  the  electric  lamp  is  particularly  rich  in  ultra-violet 
rays,  this  portion  of  its  spectrum  being  much  longer  than  in  the  case 
of  solar  light,  and  about  twice  as  long  as  the  spectrum  of  luminous 
rays.  Prisms  and  lenses  of  quartz  should  be  employed  for  this  pur- 
pose, as  this  material  is  specially  transparent  to  the  highly -refrangible 
rays.  Flint-glass  prisms,  however,  if  of  good  quality,  answer  well 
in  operating  on  solar  light.  The  luminosity  produced  by  fluorescence 
has  sensibly  the  same  tint  in  all  parts  of  the  spectrum  in  which  it 
occurs,  and  depends  upon  the  fluorescent  substance  employed.  Pris- 
matic analysis  is  not  necessary  to  the  exhibition  of  fluorescence.  The 
phenomenon  is  very  conspicuous  when  the  electric  discharge  of  a 
Holtz's  machine  or  a  RuhmkorfFs  coi-l  is  passed  near  fluorescent 
substances,  and  it  is  faintly  visible  when  these  substances  are  examined 
in  bright  sunshine.  The  light  emitted  by  a  fluorescent  substance  is 
found  by  analysis  not  to  be  homogeneous,  but  to  consist  of  rays 
having  a  wide  range  of  refrangibility. 

The  ultra-violet  rays,  though  usually  styled  invisible,  are  not 
altogether  deserving  of  this  title.  By  keeping  all  the  rest  of  the 
spectrum  out  of  sight,  and  carefully  excluding  all  extraneous  light, 
the  eye  is  enabled  to  perceive  these  highly  refrangible  rays.  Their 
colour  is  described  as  lavender-gray  or  bluish  white,  and  has  been 
attributed,  with  much  appearance  of  probability,  to  fluorescence  of 
the  retina.  The  ultra-red  rays,  on  the  other  hand,  are  never  seen; 
but  this  may  be  owing  to  the  fact,  which  has  been  established  by 
experiment,  that  they  are  largely,  if  not  entirely,  absorbed  before 
they  can  reach  the  retina. 

1052.  Recomposition  of  White  Light. — The  composite  nature  of 
white  light  can  be  established  by  actual  synthesis.  This  can  be 
done  in  several  ways. 

1.  If  a  second  prism,  precisely  similar  to  the  first,  but  with  its 
refracting  edge  turned  the  contrary  way,  is  interposed  in  the  path  of 


1068  DISPERSION.      STUDY   OF  SPECTRA. 

the  coloured  beam,  very  near  its  place  of  emergence  from  the  first 
prism,  the  deviation  produced  by  the  second  prism  will  be  equal  and 
opposite  to  that  produced  by  the  first,  the  two  prisms  will  produce 
the  effect  of  a  parallel  plate,  and  the  image  on  the  screen  will  be  a 
white  spot,  nearly  in  the  same  position  as  if  the  prisms  were 
removed. 

2.  Let  a  convex  lens  (Fig.  760)  be  interposed  in  the  path  of  the 
coloured  beam,  in  such  a  manner  that  it  receives  all  the  rays,  and 


Fig.  760.—  Recomposition  by  Lens. 

that  the  screen  and  the  prism  are  at  conjugate  focal  distances.  The 
image  thus  obtained  on  the  screen  will  be  white,  at  least  in  its  cen- 
tral portions. 

3.  Let  a  number  of  plane  mirrors  be  placed  so  as  to  receive  the 
successive  coloured  rays,  and  to  reflect  them  all  to  one  point  of  a 


Fig.  761.— Recomposition  by  Mirrors. 


screen,  as  in  Fig.  761.     The  bright  spot  thus  formed  will  be  white 
or  approximately  white. 

More  complete  information  respecting  the  mixture  of  colours  will 
be  given  in  the  next  chapter. 


SPECTROSCOPE. 


1069 


1053.  Spectroscope. — When  we  have  obtained  a  pure  spectrum  by 
any  of  the  methods  above  indicated,  we  have  in  fact  effected  an 
analysis  of  the  light  with  which  the  slit  is  illuminated.  In  recent 
years,  many  forms  of  apparatus  have  been  constructed  for  this  pur- 
pose, under  the  name  of  spectroscopes. 

A  spectroscope  usually  contains,  besides  a  slit,  a  prism,  and  a 
telescope  (as  in  Fraunhofer's  method  of  observation),  a  convex  lens 
called  a  collimator,  which  is  fixed  between  the  prism  and  the  slit, 
at  the  distance  of  its  principal  focal  length  from  the  latter.  The  effect 
of  this  arrangement  is,  that  rays  from  any  point  of  the  slit  emerge 
parallel,  as  if  they  came  from  a  much  larger  slit  (the  virtual  image 
of  the  real  slit)  at  a  much  greater  distance.  The  prism  (set  at 
minimum  deviation)  forms  a  virtual  image  of  this  image  at  the  same 
distance,  but  in  a  different  direction,  on  the  principle  of  Fig.  757. 


Fig.  762.— Spectroscope. 


To  this  second  virtual  image  the  telescope  is  directed,  being  focussed 
as  if  for  a  very  distant  object. 

Fig.  762  represents  a  spectroscope  thus  constructed.     The  tube  of 


1070  DISPERSION.      STUDY   OF  SPECTRA. 

the  collimator  is  the  further  tube  in  the  figure,  the  lens  being  at  the 
end  of  the  tube  next  the  prism,  while  at  the  far  end,  close  to  the 
lamp  flame,  there  is  a  slit  (not  visible  in  the  figure)  consisting  of  an 
opening  between  two  parallel  knife-edges,  one  of  which  can  be  moved 
to  or  from  the  other  by  turning  a  screw.  The  knife-edges  must  be 
very  true,  both  as  regards  straightness  and  parallelism,  as  it  is  often 
necessary  to  make  the  slit  exceedingly  narrow.  The  tube  on  the  left 
hand  is  the  telescope,  furnished  with  a  broad  guard  to  screen  the  eye 
from  extraneous  light.  The  near  tube,  with  a  candle  opposite  its 
end,  is  for  purposes  of  measurement.  It  contains,  at  the  end  next 
the  candle,  a  scale  of  equal  parts,  engraved  or  photographed  on  glass. 
At  the  other  end  of  the  tube  is  a  collimating  lens,  at  the  distance  of 
its  own  focal  length  from  the  scale;  and  the  collimator  is  set  so  that 
its  axis  and  the  axis  of  the  telescope  make  equal  angles  with  the 
near  face  of  the  prism.  The  observer  thus  sees  in  the  telescope,  by 
reflection  from  the  surface  of  the  prism,  a  magnified  image  of  the 
scale,  serving  as  a  standard  of  reference  for  assigning  the  positions 
of  the  lines  in  any  spectrum  which  may  be  under  examination.  This 
arrangement-  affords  great  facilities  for  rapid  observation. 

Another  plan  is,  for  the  arm  which  carries  the  telescope  to  be 
movable  round  a  graduated  circle,  the  telescope  being  furnished  with 
cross- wires,  which  the  observer  must  bring  into  coincidence  with  any 
line  whose  position  he  desires  to  measure. 

Arrangements  are  frequently  made  for  seeing  the  spectra  of  two 
different  sources  of  light  in  the  same  field  of  view,  one  half  of  the 
^  length  of  the  slit  being  illuminated  by  the  direct  rays  of 
one  of  the  sources,  while  a  reflector,  placed  opposite  the 
other  half  of  the  slit,  supplies  it  with  reflected  light 
derived   from   the  other  source.     This  method   should 
always  be  employed  when  there  is  a  question  as  to  the 
exact  coincidence  of  lines  in  the  two  spectra.     The  re- 
Fig,  res.      fleeter  is  usually  an  equilateral  prism.     The  light  enters 

Reflecting  Prism.  r  ° 

normally  at  one  of  its  faces,  is  totally  reflected  at  another, 
and  emerges  normally  at  the  third,  as  in  the  annexed  sketch  (Fig. 
763),  where  the  dotted  line  represents  the  path  of  a  ray. 

A  one-prism  spectroscope  is  amply  sufficient  for  the  ordinary  pur- 
poses of  chemistry.  For  some  astronomical  applications  a  much 
greater  dispersion  is  required.  This  is  attained  by  making  the  light 
pass  through  a  number  of  prisms  in  succession,  each  being  set  in  the 
proper  position  for  giving  minimum  deviation  to  the  rays  which  have 


COLLIMATOR. 


1071 


Fig.  764.— Train  of  Prisms. 


passed  through  its  predecessor.  Fig.  764  represents  the  ground  plan 
of  such  a  battery  of  prisms,  and  shows  the  gradually  increasing  width 
of  a  pencil  as  it  passes  round  the 
series  of  nine  prisms  on  its  way 
from  the  collimator  to  the  tele- 
scope. The  prisms  are  usually 
connected  by  a  special  arrange- 
ment, which  enables  the  observer, 
by  a  single  movement,  to  bring 
all  the  prisms  at  once  into  the 
proper  position  for  giving  mini- 
mum deviation  to  the  particular 
ray  under  examination,  a  position 
which  differs  considerably  for  rays 
of  different  refrangibilities. 

1054.  Use  of  Collimator.— The 
introduction  of  a  collimatirig  lens, 
to  be  used  in  conjunction  with  a 
prism  and  observing  telescope,  is 
due  to  Professor  Swan.1  Fraun- 

hofer  employed  no  collimator;  but  his  prism  was  at  a  distance  of 
24  feet  from  the  slit,  whereas  a  distance  of  less  than  1  foot  suffices 
when  a  collimator  is  used. 

It  is  obvious  that  homogeneous  light,  coming  from  a  point  at  the 
distance  of  a  foot,  and  falling  upon  the  whole  of  one  face  of  a  prism 
— say  an  inch  in  width,  cannot  all  have  the  incidence  proper  for 
minimum  deviation.  Those  rays  which  very  nearly  fulfil  this  con- 
dition, will  concur  in  forming  a  tolerably  sharp  image,  in  the  posi- 
tion which  we  have  already  indicated.  The  emergent  rays  taken  as 
a  whole,  do  not  diverge  from  any  one  point,  but  are  tangents  to  a 
virtual  caustic  (§  974).  An  eye  receiving  any  portion  of  these  rays, 
will  see  an  image  in  the  direction  of  a  tangent  from  the  eye  to  the 
caustic;  and  this  image  will  be  the  more  blurred  as  the  deviation  is 
further  from  the  minimum.  When  the  naked  eye  is  employed,  and 
the  prism  is  so  adjusted  that  the  centre  of  the  pupil  receives  rays  of 
minimum  deviation,  a  distance  of  five  or  six  feet  between  the  prism 
and  slit  is  sufficient  to  give  a  sharp  image;  but  if  we  employ  an 
observing  telescope  whose  object-glass  is  five  times  larger  in  diameter 
than  the  pupil  of  the  eye,  we  must  increase  the  distance  between  the 
1  Trans.  Roy.  Soc.  Edinburgh,  1847  and  1856. 


1072  DISPERSION.      STUDY   OF  SPECTRA. 

prism  and  slit  fivefold  to  obtain  equally  good  definition.  A  colli- 
mating  lens,  if  achromatic  and  of  good  quality,  gives  the  advantage 
of  good  definition  without  inconvenient  length. 

When  exact  measures  of  deviation  are  required,  it  confers  the 
further  advantage  of  altogether  dispensing  with  a  very  troublesome 
correction  for  parallax. 

1055.  Different  Kinds  of  Spectra. — The  examination  of  a  great 
variety  of  sources  of  light  has  shown  that  spectra  may  be  divided 
into  the  following  classes: — 

1.  The  solar  spectrum  is  characterized,  as  already  observed,  by  a 
definite  system  of  dark  lines  interrupting  an  otherwise  continuous 
succession  of  colours.    The  same  system  of  dark  lines  is  found  in  the 
spectra  of  the  moon  and  planets,  this  being  merely  a  consequence  of 
the  fact  that  they  shine  by  the  reflected  light  of  the  sun.   The  spectra 
of  the  fixed  stars  also  contain  systems  of  dark  lines,  which  are  different 
for  different  stars. 

2.  The  spectra  of  incandescent  solids  and  liquids  are  completely 
continuous,  containing  light  of  all  refrangibilities  from  the  extreme 
red  to  a  higher  limit  depending  on  the  temperature. 

3.  Flames  not  containing  solid  particles  in  suspension,  but  merely 
emitting  the  light  of  incandesc'ent  gases,  give  a  discontinuous  spec- 
trum, consisting  of  a  finite  number  of  bright  lines.     The  continuity 
of  the  spectrum  of  a  gas  or  candle  flame,  arises  from  the  fact  that 
nearly  all  the  light  of  the  flame  is  emitted  by  incandescent  particles 
of  solid  carbon, — particles  which  we  can  easily  collect  in  the  form  of 
soot.     When  a  gas-flame  is  fed  with  an  excessive  quantity  of  air,  as 
in  Bunsen's  burner,  the  separation  of  the  solid  particles  of  carbon 
from  the  hydrogen  with  which  they  were  combined,  no  longer  takes 
place;  the  combustion  is  purely  gaseous,  and  the  spectrum  of  the 
flame  is  found  to  consist  of  bright  lines.     When  the  electric  light  is 
produced  between  metallic  terminals,  its  spectrum  contains  bright 
lines  due  to  the  incandescent  vapour  of  these  metals,  together  with 
other  bright  lines  due  to  the  incandescence  of  the  oxygen  and  nitro- 
gen of  the  air.     When  it  is  taken  between  charcoal  terminals,  its 
spectrum  is  continuous;   but  if  metallic  particles  be  present,  the 
bright  lines  due  to  their  vapours  can  be  seen  as  well. 

The  spectrum  of  the  electric  discharge  in  a  Geissler's  tube  consists 
of  bright  lines  characteristic  of  the  gas  contained  in  the  tube. 

1056.  Spectrum  Analysis. — As  the  spectrum  exhibited  by  a  com- 
pound substance  when  subjected  to  the  action  of  heat,  is  frequently 


SPECTRA    OF    VARIOUS    SOURCES    OF  LIGHT 


PLATE  111. 


7?ie  Sun,  2. 77ie  Sbrisedge.  '6.  Sodiwn-A.  Potassium.  5.  litiwun.  6.  Caesium*  IRuhidJwn-  8  ThaJlvum 
Q-Qzldum.  \&Sti-OTutiuri-t,\lBariu}fi,.\?..IruiiM.rrL  IT^.Phosphoru.s.  14 Hydrogen. 


SPECTRUM  ANALYSIS.  1073 

found  to  be  identical  with  the  spectrum  of  one  of  its  constituents,  or 
to  consist  of  the  spectra  of  its  constituents  superimposed,1  the  spec- 
troscope affords  an  exceedingly  ready  method  of  performing  qualita- 
tive analysis. 

If  a  salt  of  a  metal  which  is  easily  volatilized  is  introduced  into  a 
Bunsen  lamp-flame,  by  means  of  a  loop  of  platinum  wire,  the  bright 
lines  which  form  the  spectrum  of  the  metal  will  at  once  be  seen 
in  a  spectroscope  directed  to  the  flame;  and  the  spectrum  of  the 
Bunsen  flame  itself  is  too  faint  to  introduce  any  confusion.  For 
those  metals  which  require  a  higher  temperature  to  volatilize  them, 
electric  discharge  is  usually  employed.  Geissler's  tubes  are  com- 
monly used  for  gases. 

Plate  III.  contains  representations  of  the  spectra  of  several  of  the 
more  easily  volatilized  metals,  as  well  as  of  phosphorus  and  hydro- 
gen; and  the  solar  spectrum  is  given  at  the  top  for  comparison. 
The  bright  lines  of  some  of  these  substances  are  precisely  coincident 
with  some  of  the  dark  lines  in  the  solar  spectrum. 

The  fact  that  certain  substances  when  incandescent  give  definite 
bright  lines,  has  been  known  for  many  years,  from  the  researches  of 
Brewster,  Herschel,  Talbot,  and  others;  but  it  was  for  a  long  time 
thought  that  the  same  line  might  be  produced  by  different  sub- 
stances, more  especially  as  the  bright  yellow  line  of  sodium  was 
often  seen  in  flames  in  which  that  metal  was  not  supposed  to  be 
present.  Professor  Swan,  having  ascertained  that  the  presence  of 
the  2,500,000th  part  of  a  grain  of  sodium  in  a  flame  was  sufficient 
to  produce  it,  considered  himself  justified  in  asserting,  in  1856,  that 
this  line  was  always  to  be  taken  as  an  indication  of  the  presence  of 
sodium  in  larger  or  smaller  quantity. 

But  the  greatest  advance  in  spectral  analysis  was  made  by  Bunsen 
and  Kirchhoff,  who,  by  means  of  a  four-prism  spectroscope,  obtained 
accurate  observations  of  the  positions  of  the  bright  lines  in  the 
spectra  of  a  great  number  of  substances,  as  well  as  of  the  dark  lines 
in  the  solar  spectrum,  and  called  attention  to  the  identity  of  several 
of  the  latter  with  several  of  the  former.  Since  the  publication  of 
their  researches,  the  spectroscope  has  come  into  general  use  among 
chemists,  and  has  already  led  to  the  discovery  of  four  new  metals, 
csesium,  rubidium,  thallium,  and  indium. 

1057.  Reversal  of  Bright  Lines.     Analysis  of  the  Sun's  Atmosphere. 

1  These  appear  to  be  merely  examples  of  the  dissociation  of  the  elements  of  a  chemical 
compound  at  high  temperatures. 
68 


1074  DISPERSION.      STUDY   OF   SPECTRA. 

— It  may  seem  surprising  that,  while  incandescent  solids  and  liquids 
are  found  to  give  continuous  spectra  containing  rays  of  all  refran- 
gibilities,  the  solar  spectrum  is  interrupted  by  dark  lines  indicating 
the  absence  or  relative  feebleness  of  certain  elementary  rays.  It 
seems  natural  to  suppose  that  the  deficient  rays  have  been  removed  by 
selective  absorption,  and  this  conjecture  was  thrown  out  long  since. 
But  where  and  how  is  this  absorption  produced  ?  These  questions 
have  now  received  an  answer  which  appears  completely  satisfactory. 

According  to  the  theory  of  exchanges,  which  has  been  explained 
in  connection  with  the  radiation  of  heat  (§  464, 483),  every  substance 
which  emits  certain  kinds  of  rays  to  the  exclusion  of  others,  absorbs 
the  same  kind  which  it  emits;  and  when  its  temperature  is  the  same 
in  the  two  cases  compared,  its  emissive  and  absorbing  power  are  pre- 
cisely equal  for  any  one  elementary  ray. 

When  an  incandescent  vapour,  emitting  only  rays  of  certain 
definite  refrangibilities,  and  therefore  having  a  spectrum  of  bright 
lines,  is  interposed  between  the  observer  and  a  very  bright  source 
of  light,  giving  a  continuous  spectrum,  the  vapour  allows  no  rays  of 
its  own  peculiar  kinds  to  pass;  so  that  the  light  which  actually  comes 
to  the  observer  consists  of  transmitted  rays  in  which  these  particular 
kinds  are  wanting,  together  with  the  rays  emitted  by  the  vapour 
itself,  these  latter  being  of  precisely  the  same  kind  as  those  which  it 
has  refused  to  transmit.  It  depends  on  the  relative  brightness  of  the 
two  sources  whether  these  particular  rays  shall  be  on  the  whole  in 
excess  or  defect  as  compared  with  the  rest.  If  the  two  sources  are 
at  all  comparable  in  brightness,  these  rays  will  be  greatly  in  excess, 
inasmuch  as  they  constitute  the  whole  light  of  the  one,  and  only  a 
minute  fraction  of  the  light  of  the  other;  but  the  light  of  the  electric 
lamp,  or  of  the  lime-light,  is  usually  found  sufficiently  powerful  to 
produce  the  contrary  effect;  so  that  if,  for  example,  a  spirit-lamp  with 
salted  wick  is  interposed  between  the  slit  of  a  spectroscope  and  the 
electric  light,  the  bright  yellow  line  due  to  the  sodium  appears  black 
by  contrast  with  the  much  brighter  back-ground  which  belongs  to 
the  continuous  spectrum  of  the  charcoal  points.  By  employing  only 
some  10  or  15  cells,  a  light  may  be  obtained,  the  yellow  portion  of 
which,  as  seen  in  a  one-prism  spectroscope,  is  sensibly  equal  in 
brightness  to  the  yellow  line  of  the  sodium  flame,  so  that  this  line 
can  no  longer  be  separately  detected,  and  the  appearance  is  the  same 
whether  the  sodium  flame  be  interposed  or  removed. 

The  dark  lines  in  the  solar  spectrum  would  therefore  be  accounted 


CHROMOSPHERE  AND   PHOTOSPHERE.  1075 

for  by  supposing  that  the  principal  portion  of  the  sun's  light  conies 
from  an  inner  stratum  which  gives  a  continuous  spectrum,  and  that 
a  layer  external  to  this  contains  vapours  which  absorb  particular 
rays,  and  thus  produce  the  dark  lines.  The  stratum  which  gives 
the  continuous  spectrum  might  be  solid,  liquid,  or  even  gaseous,  for 
the  experiments  of  Frankland  and  Lockyer  have  shown  that,  as  the 
pressure  of  a  gas  is  increased,  its  bright  lines  broaden  out  into  bands, 
and  that  the  bands  at  length  become  so  wide  as  to  join  each  other 
and  form  a  continuous  spectrum.1 

Hydrogen,  sodium,  calcium,  barium,  magnesium,  zinc,  iron,  chro- 
mium, cobalt,  nickel,  copper,  and  manganese  have  all  been  proved 
to  exist  in  the  sun  by  the  accurate  identity  of  position  of  their 
bright  lines  with  certain  dark  lines  in  the  sun's  spectrum. 

The  strong  line  D,  which  in  a  good  instrument  is  seen  to  consist 
of  two  lines  near  together,  is  due  to  sodium;  and  the  lines  C  and  F 
are  due  to  hydrogen.  No  less  than  450  of  the  solar  dark  lines  have 
been  identified  with  bright  lines  of  iron. 

1058.  Telespectroscope.  Solar  Prominences. — For  astronomical  in- 
vestigations, the  spectroscope  is  usually  fitted  to  a  telescope,  and  takes 
the  place  of  the  eye-piece,  the  plane  of  the  slit  being  placed  in  the 
principal  focus  of  the  object-glass,  so  that  the  image  is  thrown  upon 
it,  and  the  light  which  enters  the  slit  is  the  light  which  forms  one 
strip  (so  to  speak)  of  the  image,  and  which  therefore  comes  from  one 
strip  of  the  object.  A  telescope  thus  equipped  is  called  a  telespectro- 
scope.  Extremely  interesting  results  have  been  obtained  by  thus 
subjecting  to  examination  a  strip  of  the  sun's  edge,  the  strip  being 
sometimes  tangential  to  the  sun's  disc,  and  sometimes  radial.  When 
the  former  arrangement  is  adopted,  the  appearance  presented  is  that 
depicted  in  No.  2,  Plate  III.,  consisting  of  a  few  bright  lines  scattered 
through  a  back-ground  of  the  ordinary  solar  spectrum.  The  bright 
lines  are  due  to  an  outer  layer  called  the  chromosphere,  which 
is  thus  proved  to  be  vaporous.  The  ordinary  solar  spectrum  which 
accompanies  it,  is  due  to  that  part  of  the  sun  from  which  most 
of  our  light  is  derived.  This  part  is  called  the  photosphere,  and  if 
not  solid  or  liquid,  it  must  consist  of  vapour  so  highly  compressed 
that  its  properties  approximate  to  those  of  a  liquid. 

When  the  slit  is  placed  radially,  in  such  a  position  that  only  a 

1  The  gradual  transition  from  a  spectrum  of  bright  lines  to  a  continuous  spectrum  may 
be  held  to  be  an  illustration  of  the  continuous  transition  which  can  be  effected  from  the 
condition  of  ordinary  gas  to  that  of  ordinary  liquid  (§  380). 


1076  DISPERSION.      STUDY   OF  SPECTRA. 

small  portion  of  its  length  receives  light  from  the  body  of  the  sun, 
the  spectra  of  the  photosphere  and  chromosphere  are  seen  in  imme- 
diate contiguity,  and  the  bright  lines  in  the  latter  (notably  those  of 
hydrogen,  No.  14,  Plate  III.)  are  observed  to  form  continuations  of 
some  of  the  dark  lines  of  the  former. 

The  chromosphere  is  so  much  less  bright  than  the  photosphere, 
that,  until  a  few  years  since,  its  existence  was  never  revealed  except 
during  total  eclipses  of  the  sun,  when  projecting  portions  of  it  were 
seen  extending  beyond  the  dark  body  of  the  moon.  The  spectrum 
of  these  projecting  portions,  which  have  been  variously  called 
"prominences,"  "red  flames,"  and  "rose-coloured  protuberances,"  was 
first  observed  during  the  "Indian  eclipse"  of  1868,  and  was  found  to 
consist  of  bright  lines,  including  those  of  hydrogen.  From  their 
excessive  brightness,  M.  Janssen,  who  was  one  of  the  observers, 
expressed  confidence  that  he  should  be  able  to  see  them  in  full 
sunshine;  and  the  same  idea  had  been  already  conceived  and  pub- 
lished by  Mr.  Lockyer.  The  expectation  was  shortly  afterwards 
realized  by  both  these  observers,  and  the  chromosphere  has  ever 
since  been  an  object  of  frequent  observation.  The  visibility  of  the 
chromosphere  lines  in  full  sunshine,  depends  upon  the  principle  that, 
while  a  continupus  spectrum  is  extended,  and  therefore  made  fainter, 
by  increased  dispersion,  a  bright  line  in  a  spectrum  is  not  sensibly 
broadened,  and  therefore  loses  very  little  of  its  intrinsic  brightness 
(§  1061).  Very  high  dispersion  is  necessary  for  this  purpose. 

Still  more  recently,  by  opening  the  slit  to  about  the  average  width 
of  the  prominence-region,  as  measured  on  the  image  of  the  sun  which 
is  thrown  on  the  slit,  it  has  been  found  possible  to  see  the  whole  of 
an  average-sized  prominence  at  one. view.  This  will  be  understood 
by  remembering  that  a  bright  line  as  seen  in  a  spectrum  is  a  mono- 
chromatic image  of  the  illuminated  portion  of  the  slit,  or  when  a  tele- 
spectroscope  is  used,  as  in  the  present  case,  it  is  a  monochromatic 
image  of  one  strip  of  the  image  formed  by  the  object-glass,  namely, 
that  strip  which  coincides  with  the  slit.  If  this  strip  then  contains 
a  prominence  in  which  the  elementary  rays  C  and  F  (No.  2,  Plate  III.) 
are  much  stronger  than  in  the  rest  of  the  strip,  a  red  image  of  the 
prominence  will  be  seen  in  the  part  of  the  spectrum  corresponding 
to  the  line  C,  and  a  blue  image  in  the  place  corresponding  to  the 
line  F.  This  method  of  observation  requires  greater  dispersion  than 
is  necessary  for  the  mere  detection  of  the  chromosphere  lines;  the 
dispersion  required  for  enabling  a  bright-line  spectrum  to  predomi- 


DOPPLER'S  PRINCIPLE.  1077 

nate  over  a  continuous  spectrum  being  always  nearly  proportional 
to  the  width  of  the  slit  (§  1061). 

Of  the  nebulae,  it  is  well  known  that  some  have  been  resolved  by 
powerful  telescopes  into  clusters  of  stars,  while  others  have  as  yet 
proved  irresolvable.  Huggins  has  found  that  the  former  class  of 
nebulae  give  spectra  of  the  same  general  character  as  the  sun  and  the 
fixed  stars,  but  that  some  of  the  latter  class  give  spectra  of  bright 
lines,  indicating  that  their  constitution  is  gaseous. 

1059.  Displacement  of  Lines  consequent  on  Celestial  Motions. — 
According  to  the  undulatory  theory  of  light,  which  is  now  univer- 
sally accepted,  the  fundamental  difference  between  the  different  rays 
which  compose  the  complete  spectrum,  is  a  difference  of  wave- 
frequency,  and,  as  connected  with  this,  a  difference  of  wave-length 
in  any  given  medium,  the  rays  of  greatest  wave-frequency  or  shortest 
wave-length  being  the  most  refrangible. 

Doppler  first  called  attention  to  the  change  of  refrangibility  which 
must  be  expected  to  ensue  from  the  mutual  approach  or  recess  of  the 
observer  and  the  source  of  light,  the  expectation  being  grounded  on 
reasoning  which  we  have  explained  in  connection  with  acoustics 
(§  898). 

Doppler  adduced  this  principle  to  explain  the  colours  of  the  fixed 
stars,  a  purpose  to  which  it  is  quite  inadequate;  but  it  has  rendered 
very  important  service  in  connection  with  spectroscopic  research. 
Displacement  of  a  line  towards  the  more  refrangible  end  of  the  spec- 
trum, indicates  approach,  displacement  in  the  opposite  direction  indi- 
cates recess,  and  the  velocity  of  approach  or  recess  admits  of  being 
calculated  from  the  observed  displacement. 

When  the  slit  of  the  spectroscope  crosses  a  spot  on  the  sun's  disc, 
the  dark  lines  lose  their  straightness  in  this  part,  and  are  bent,  some- 
times to  one  side,  sometimes  to  the  other.  These  appearances  clearly 
indicate  uprush  and  downrush  of  gases  in  the  sun's  atmosphere  in 
the  region  occupied  by  the  Spot. 

Huggins  detected  a  displacement  of  the  F  line  towards  the  red 
end,  in  the  spectrum  of  Sirius,  as  compared  with  the  spectrum  of  the 
sun  or  of  hydrogen.  The  displacement  is  so  small  as  only  to  admit 
of  measurement  by  very  powerful  instrumental  appliances;  but, 
small  as  it  is,  calculation  shows  that  it  indicates  a  motion  of  recess 
at  the  rate  of  about  30  miles  per  second.1 

1  The  observed  displacement  corresponded  to  recess  at  the  rate  of  41 '4  miles  per  second; 
but  12-0  of  this  must  be  deducted  for  the  motion  of  the  earth  in  its  orbit  at  the  season  of 


1078  •  DISPERSION.      STUDY  OF   SPECTRA. 

1060.  Spectra  of  Artificial  Lights.— The  spectra  of  the  artificial 
lights  in  ordinary  use  (including  gas,  oil-lamps,  and  candles)  differ 
from  the  solar  spectrum  in  the  relative  brightness  of  the  different 
colours,  as  well  as  in  the  entire  absence  of  dark  lines.     They  are 
comparatively  strong  in  red  and  green,  but  weak  in  blue;  hence  all 
colours  which  contain  much  blue  in  their  composition  appear  to 
disadvantage  by  gas-light. 

It  is  possible  to  find  artificial  lights  whose  spectra  are  of  a  com- 
pletely different  character.  The  salts  of  strontium,  for  example,  give 
red  light,  composed  of  the  ingredients  represented  in  spectrum 
No.  10,  Plate  III.,  and  those  of  sodium  yellow  light  (No.  3,  Plate  III.). 
If  a  room  is  illuminated  by  a  sodium  flame  (for  example,  by  a  spirit- 
lamp  with  salt  sprinkled  on  the  wick),  all  objects  in  the  room  will 
appear  of  a  uniform  colour  (that  of  the  flame  itself),  differing  only 
in  brightness,  those  which  contain  no  yellow  in  their  spectrum  as 
seen  by  day-light  being  changed  to  black.  The  human  countenance 
and  hands  assume  a  ghastly  hue,  and  the  lips  are  no  longer 
red. 

A  similar  phenomenon  is  observed  when  a  coloured  body  is  held 
in  different  parts  of  the  solar  spectrum  in  a  dark  room,  so  as  to  be 
illuminated  by  different  kinds  of  monochromatic  light.  The  object 
either  appears  of  the  same  colour  as  the  light  which  falls  upon  it,  or 
else  it  refuses  to  reflect  this  light  and  appears  black.  Hence  a  screen 
for  exhibiting  the  spectrum  should  be  white. 

1061.  Brightness  and  Purity. — The  laws  which  determine  the  bright- 
ness of  images  generally,  and  which  have  been  expounded  at  some 
length  in  the  preceding  chapter,  may  be  applied  to  the  spectroscope. 
We  shall,  in  the  first  instance,  neglect  the  loss  of  light  by  reflection 
and  imperfect  transmission. 

Let  A  denote  the  prismatic  dispersion,  as  measured  by  the  angular 
separation  of  two  specified  monochromatic  images  when  the  naked 
eye  is  applied  to  the  last  prism,  the  observing  telescope  being  re- 
moved. Then,  putting  ra  for  the  linear  magnifying  power  of  the 

the  year  when  the  observation  was  made.     The  remainder,  29'4,  was  therefore  the  rate  at 
which  the  distance  between  the  sun  and  Sirius  was  increasing. 

In  a  more  recent  paper  Dr.  Huggins  gave  the  results  of  observations  with  more  powerful 
instrumental  appliances.  The  recess  of  Sirius  was  found  to  be  only  20  miles  per  second. 
Arcturus  was  found  to  be  approaching  at  the  rate  of  50  miles  per  second.  Community 
of  motion  was  established  in  certain  sets  of  stars;  and  the  belief  previously  held  by 
astronomers,  as  to  the  direction  in  which  the  solar  system  is  moving  with  respect  to  the 
stars  as  a  whole,  was  fully  confirmed. 


BRIGHTNESS  AND   PURITY   OF   SPECTRUM.  1079 

telescope,  m  A  is  the  angular  separation  observed  when  the  eye  is 
applied  to  the  telescope.     We  shall  call  m  A  the  total  dispersion. 

Let  0  denote  the  angle  which  the  breadth  of  the  slit  subtends 
at  the  centre  of  the  collimating  lens,  and  which  is  measured  by 

Then  e  is  also  the  apparent  breadth  of  any  absolutely 


monochromatic  image  of  the  slit,  formed  by  rays  of  minimum  devia- 
tion, as  seen  by  an  eye  applied  either  to  the  first  prism,  the  last 
prism,  or  any  one  of  the  train  of  prisms.  The  change  produced  in 
a  pencil  of  monochromatic  rays  by  transmission  through  a  prism  at 
minimum  deviation,  is  in  fact  simply  a  change  of  direction,  without 
any  change  of  mutual  inclination;  and  thus  neither  brightness  nor 
apparent  size  is  at  all  affected.  In  ordinary  cases,  the  bright  lines  of 
a  spectrum  may  be  regarded  as  monochromatic,  and  their  apparent 
breadth,  as  seen  without  the  telescope,  is  sensibly  equal  to  0.  Strictly 
speaking,  the  effect  of  prismatic  dispersion  in  actual  cases,  is  to 
increase  the  apparent  breadth  by  a  small  quantity,  which,  if  all  the 
prisms  are  alike,  is  proportional  to  the  number  of  prisms;  but  the 
increase  is  usually  too  small  to  be  sensible. 

Let  I  denote  the  intrinsic  brightness  of  the  source  as  regards  any 
one  of  its  (approximately)  monochromatic  constituents;  in  other 
words,  the  brightness  which  the  source  would  have  if  deprived  of  all 
its  light  except  that  which  goes  to  form  a  particular  bright  line. 
Then,  still  neglecting  the  light  stopped  by  the  instrument,  the  bright- 
ness of  this  line  as  seen  without  the  aid  of  the  telescope  will  be  I; 
and  as  seen  in  the  telescope  it  will  either  be  equal  to  or  less  than 
this,  according  to  the  magnifying  power  of  the  telescope  and  the 
effective  aperture  of  the  object-glass  (§  1038).  If  the  breadth  of  the 
slit  be  halved,  the  breadth  of  the  bright  line  will  be  halved,  and  its 
brightness  will  be  unchanged.  These  conclusions  remain  true  so  long 
as  the  bright  line  can  be  regarded  as  practically  monochromatic. 

The  brightness  of  any  part  of  a  continuous  spectrum  follows  a 
very  different  law.  It  varies  directly  as  the  width  of  the  slit,  and 
inversely  as  the  prismatic  dispersion.  Its  value  without  the  observ- 

ing telescope,  or  its  maximum  value  with  a  telescope,  is  —  i,  where  i 
is  a  coefficient  depending  only  on  the  source. 

The  purity  of  any  part  of  a  continuous  spectrum  is  properly 
measured  by  the  ratio  of  the  distance  between  two  specified  mono- 
chromatic images  to  the  breadth  of  either,  the  distance  in  question 
being  measured  from  the  centre  of  one  to  the  centre  of  the  other. 


1080  DISPERSION.      STUDY  OF   SPECTRA. 

This  ratio  is  unaffected  by  the  employment  of  an  observing  telescope, 
and  is  -J. 

The  ratio  of  the  brightness  of  a  bright  line  to  that  of  the  adjacent 
portion  of  a  continuous  spectrum  forming  its  back -ground,  is  -^j> 

assuming  the  line  to  be  so  nearly  monochromatic  that  the  increase 
of  its  breadth  produced  by  the  dispersion  of  the  prisms  is  an  insigni- 
ficant fraction  of  its  whole  breadth.  As  we  widen  the  slit,  and  so 
increase  0,  we  must  increase  A  in  the  same  ratio,  if  we  wish  to 
preserve  the  same  ratio  of  brightness.  As  *  is  increased  indefinitely, 
the  predominance  of  the  bright  lines  does  not  increase  indefinitely, 
but  tends  to  a  definite  limit,  namely,  to  the  predominance  which 
they  would  have  in  a  perfectly  pure  spectrum  of  the  given  source. 

The  loss  of  light  by  reflection  and  imperfect  transmission,  increases 
with  the  number  of  surfaces  of  glass  which  are  to  be  traversed;  so 
that,  with  a  long  train  of  prisms  and  an  observing  telescope,  the 
actual  brightness  will  always  be  much  less  than  the  theoretical 
brightness  as  above  computed. 

The  actual  purity  is  always  less  than  the  theoretical  purity,  being 
greatly  dependent  on  freedom  from  optical  imperfections;  and  these 
can  be  much  more  completely  avoided  in  lenses  than  in  prisms.  It 
is  said  that  a  single  good  prism,  with  a  first-class  collimator  and 
telescope  (as  originally  employed  by  Swan),  gives  a  spectrum  much 
more  free  from  blurring  than  the  modern  multiprism  spectroscopes, 
when  the  total  dispersion  m  A  is  the  same  in  both  the  cases  com- 
pared. 

1062.  Chromatic  Aberration. —  The  unequal  rtfrangibility  of  the 
different  elementary  rays  is  a  source  of  grave  inconvenience  in  con- 
nection with  lenses.  The  focal  length  of  a  lens  depends  upon  its 
index  of  refraction,  which  of  course  increases  with  refrangibility,  the 
focal  length  being  shortest  for  the  most  refrangible  rays.  Thus  a 
lens  of  uniform  material  will  not  form  a  single  white  image  of  a 
white  object,  but  a  series  of  images,  of  all  the  colours  of  the  spectrum, 
arranged  at  different  distances,  the  violet  images  being  nearest,  and 
the  red  most  remote.  If  we  place  a  screen  anywhere  in  the  series 
of  images,  it  can  only  be  in  the  right  position  for  one  colour.  Every 
other  colour  will  give  a  blurred  image,  and  the  superposition  of  them 
all  produces  the  image  actually  formed  on  the  screen.  If  the  object 
be  a  uniform  white  spot  on  a  black  ground,  its  image  on  the  screen 


ACHEOMATISM.  1081 

will  consist  of  white  in  its  central  parts,  gradually  merging  into  a 
coloured  fringe  at  its  edge.  Sharpness  of  outline  is  thus  rendered 
impossible,  and  nothing  better  can  be  done  than  to  place  the  screen 
at  the  focal  distance  corresponding  to  the  brightest  part  of  the  spec- 
trum. Similar  indistinctness  will  attach  to  images  observed  in  mid- 
air, whether  directly  or  by  means  of  another  lens.  This  source  of 
confusion  is  called  chromatic  aberration. 

1063.  Possibility  of  Achromatism. — In  order  to  ascertain  whether 
it  was  possible  to  remedy  this  evil   by  combining  lenses  of   two 
different  materials,  Newton  made  some  trials  with  a  compound  prism 
composed  of  glass  and  water  (the  latter  containing  a  little  sugar  of 
lead),  and  he  found  that  it  was  not  possible,  by  any  arrangement  of 
these  two  substances,  to  produce  deviation  of  the  transmitted  light 
without  separation  into  its  component  colours.      Unfortunately  he 
did  not  extend  his  trials  to  other  substances,  but  concluded  at  once 
that  an  achromatic  prism  (and  hence  also  an  achromatic  lens)  was 
an  impossibility;  and  this  conclusion  was  for  a  long  time  accepted 
as  indisputable.     Mr.  Hall,  a  gentleman  of  Worcestershire,  was  the 
first  to  show  that  it  was  erroneous,  and  is  said  to  have  constructed 
some  achromatic  telescopes;  but  the  important  fact  thus  discovered 
did  not  become  generally  known  till  it  was  rediscovered  by  Dollond, 
an  eminent  London  optician,  in  whose  hands  the  manufacture  of 
achromatic  instruments  attained  great  perfection. 

1064.  Conditions  of  Achromatism. — The  conditions  necessary  for 
achromatism  are  easily  explained.     The  angular  separation  between 
the  brightest  red  and  the  brightest  violet  ray  transmitted  through  a 
prism  is  called  the  dispersion  of  the  prism,  and  is  evidently  the 
difference  of  the  deviations  of  these  rays.     These  deviations,  for  the 
position  of  minimum  deviation  of  a  prism  of  small  refracting  angle 
A,  are  (//  —  1)  A  and  (//'  —  1)  A,  p!  and  /«"  denoting  the  indices  of 
refraction  for  the  two  rays  considered  (§  1004,  equation  (1))  and 
their  difference  is  (//  -  /*')  A.     This  difference  is  always  small  in 
comparison  with  either  of  the  deviations  whose  difference  it  is,  and 
its  ratio  to  either  of  them,  or  more  accurately  its  ratio  to  the  value 
of  (n  —  1)  A  for  the  brightest  part  of  the  spectrum,  is  called  the  dis- 
'persive  power  of  the  substance.     As  the  common  factor  A  may  be 

omitted,  the  formula  for  the  dispersive  power  is  evidently  ^-£y- 

If  this  ratio  were  the  same  for  all  substances,  as  Newton  supposed, 
achromatism  would  be  impossible;  but  in  fact  its  value  varies  greatly, 


1082  DISPEKSION.      STUDY   OF   SPECTRA. 

and  is  greater  for  flint  than  for  crown  glass.  If  two  prisms  of  these 
substances,  of  small  refracting  angles,  be  combined  into  one,  with 
their  edges  turned  opposite  ways,  they  will  achromatize  one  another 
if  (/«"  -  (£«')  A,  or  the  product  of  deviation  by  dispersive  power,  is  the 
same  for  both.  As  the  deviations  can  be  made  to  have  any  ratio  we 
please  by  altering  the  angles  of  the  prisms,  the  condition  is  evidently 
possible. 

The  deviation  which  a  simple  ray  undergoes  in  traversing  a  lens, 

at  a  distance  x  from  the  axis,  is  4-, /denoting  the  focal  length  of  the 

lens  (§  1004),  and  the  separation  of  the  red  and  violet  constituents 
of  a  compound  ray  is  the  product  of  this  deviation  by  the  dispersive 
power  of  the  material.  If  a  convex  and  concave  lens  are  combined, 
fitting  closely  together,  the  deviations  which  they  produce  in  a  ray 
traversing  both,  are  in  opposite  directions,  and  so  also  are  the  dis- 
persions. If  we  may  regard  x  as  having  the  same  value  for  both  (a 
supposition  which  amounts  to  neglecting  the  thicknesses  of  the  lenses 
in  comparison  with  their  focal  lengths)  the  condition  of  no  resultant 
dispersion  is  that  j 

dispersive  power  x   -„- 

has  the  same  value  for  both  lenses.  Their  focal  lengths  must  there- 
fore be  directly  as  the  dispersive  powers  of  their  materials.  These 
latter  are  about  '033  for  crown  and  '052  for  flint  glass.  A  converg- 
ing achromatic  lens  usually  consists  of  a  double  convex  lens  of  crown 
fitted  to  a  diverging  meniscus  of  flint.  In  every  achromatic  com- 
bination of  two  pieces,  the  direction  of  resultant  deviation  is  that 
due  to  the  piece  of  smaller  dispersive  power. 

The  definition  above  given  of  dispersive  power  is  rather  loose. 
To  make  it  accurate,  we  must  specify,  by  reference  to  the  "  fixed 
lines,"  the  precise  positions  of  the  two  rays  whose  separation  we 
consider. 

Since  the  distances  between  the  fixed  lines  have  different  propor- 
tions for  crown  and  flint  glass,  achromatism  of  the  whole  spectrum  is 
impossible.  With  two  pieces  it  is  possible  to  unite  any  two  selected 
rays,  with  three  pieces  any  three  selected  rays,  and  so  on.  It  is 
considered  a  sign  of  good  achromatism  when  no  colours  can  be 
brought  into  view  by  bad  focussing  except  purple  and  green. 

1065.  Achromatic  Eye-pieces. — The  eye-pieces  of  microscopes  and 
astronomical  telescopes,  usually  consist  of  two  lenses  of  the  same  kind 
of  glass,  so  arranged  as  to  counteract,  to  some  extent,  the  spherical 


RAINBOW. 


1083 


and  chromatic  aberrations  of  the  object-glass.  The  positive  eye-piece, 
invented  by  Ramsden,  is  suited  for  observation  with  cross-wires  or 
micrometers;  the  negative  eye-piece,  invented  by  Huygens,  is  not 
adapted  for  purposes  of  measurement,  but  is  preferred  when  distinct 
vision  is  the  sole  requisite.  These  eye-pieces  are  commonly  called 
achromatic,  but  their  achromatism  is  in  a  manner  spurious.  It  con- 
sists not  in  bringing  the  red  and  violet  images  into  true  coincidence, 
but  merely  in  causing  one  to  cover  the  other  as  seen  from  the  posi- 
tion occupied  by  the  observer's  eye. 

In  the  best  opera-glasses  (§  1033),  the  eye-piece,  as  well  as  the 
object-glass,  is  composed  of  lenses  of  flint  and  crown  so  combined  as 
to  be  achromatic  in  the  more  proper  sense  of  the  word. 

1066.  Rainbow. — The  unequal  refrangibility  of  the  different  ele- 
mentary rays  furnishes  a  complete  explanation  of  the  ordinary 
phenomena  of  rainbows.  The  explanation  was  first  given  by  New- 
ton, who  confirmed  it  by  actual  measurement. 

It  is  well  known  that  rainbows  are  seen  when  the  sun  is  shining 
on  drops  of  water.  Sometimes  one  bow  is  seen,  sometimes  two, 
each  of  them  presenting  colours  resembling  those  of  the  solar 
spectrum.  When  there  is  only  one  bow,  the  red  arch  is  above 
and  the  violet  below.  When  there  is  a  second  bow,  it  is  at  some 
distance  outside  of  this,  has  the  colours  in  reverse  order,  and  is 
usually  less  bright. 

Rainbows  are  often  observed  in  the  spray  of  cascades  and  fountains, 
when  the  sun  is  shining. 

In  every  case,  a  line  join- 
ing the  observer  to  the  sun 
is  the  axis  of  the  bow  or 
bows;  that  is  to  say,  all 
parts  of  the  length  of  the 
bow  are  at  the  same  angu- 
lar distance  from  the  sun. 

The  formation  of  the  pri- 
mary bow  is  illustrated  by 
Fig.  765.  A  ray  of  solar 
light,  falling  on  a  spherical 
drop  of  water,  in  the  direc- 
tion S  I,  is  refracted  at  I, 
then  reflected  internally 
from  the  back  of  the  drop,  and  again  refracted  into  the  air  in  the 


Fig.  765.— Production  of  Primary  Bow. 


1084  DISPERSION.      STUDY  OF  SPECTRA. 

direction  I'  M.  If  we  take  different  points  of  incidence,  we  shall 
obtain  different  directions  of  emergence,  so  that  the  whole  light 
which  emerges  from  the  drop  after  undergoing,  as  in  the  figure, 
two  refractions  and  one  reflection,  forms  a  widely-divergent  pencil. 
Some  portions  of  this  pencil,  however,  contain  very  little  light.  This 
is  especially  the  case  with  those  rays  which,  having  been  incident 
nearly  normally,  are  returned  almost  directly  back,  and  also  with 
those  which  were  almost  tangential  at  incidence.  The  greatest  con- 
densation, as  regards  any  particular  species  of  elementary  ray,  occurs 
at  that  part  of  the  emergent  pencil  which  has  undergone  minimum 
deviation.  It  is  by  means  of  rays  which  have  undergone  this  mini- 
mum deviation,  that  the  observer  sees  the  corresponding  colour  in 
the  bow;  and  the  deviation  which  they  have  undergone  is  evidently 
equal  to  the  angular  distance  of  this  part  of  the  bow  from  the  sun. 

The  minimum  deviation  will  be  greatest  for  those  rays  which  are 
most  refrangible.  If  the  figure,  for  example,  be  supposed  to  represent 
the  circumstances  of  minimum  deviation  for  violet,  we  shall  obtain 
smaller  deviation  in  the  case  of  red,  even  by  giving  the  angle  I A  I' 
the  same  value  which  it  has  in  the  case  of  minimum  deviation  for 
violet,  and  still  more  when  we  give  it  the  value  which  corresponds 
to  the  minimum  deviation  of  red.  The  most  refrangible  colours  are 
accordingly  seen  furthest  from  the  sun.  The  effect  of  the  rays  which 
undergo  other  than  minimum  deviation,  is  to  produce  a  border  of 
white  light  on  the  side  remote  from  the  sun;  that  is  to  say,  on  the 
inner  edge  of  the  bow.1 

1  When  the  drops  are  very  uniform  in  size,  a  series  of  faint  supernumerary  bou'S,  alter- 
nately pnrple  and  green,  is  sometimes  seen  beneath  the  primary  bow.  These  bows  are 
produced  by  the  mutual  interference  of  rays  which  have  undergone  other  than  minimum 
deviation,  and  the  interference  arises  in  the  following  way.  Any  two  parallel  directions 
of  emergence,  for  rays  of  a  given  refrangibility,  correspond  in  general  to  two  different 
points  of  incidence  on  any  given  drop,  one  of  the  two  incident  rays  being  more  nearly 
normal,  and  the  other  more  nearly  tangential  to  the  drop  than  the  ray  of  minimum 
deviation.  These  two  rays  have  pursued  dissimilar  paths  in  the  drop,  and  are  in  diiferent 
phases  when  they  reach  the  observer's  eye.  The  difference  of  phase  may  amount  to  one, 
two,  three,  or  more  exact  wave-lengths,  and  thus  one,  two,  three,  or  more  supernumerary 
bows  may  be  formed.  The  distances  between  the  supernumerary  bows  will  be  greater  as 
the  drops  of  water  are  smaller.  This  explanation  is  due  to  Dr.  Thomas  Young. 

A  more  complete  theory,  in  which  diffraction  is  taken  into  account,  is  given  by  Airy 
in  the  Cambridge  Transactions  for  1838;  and  the  volume  for  the  following  year  contains 
an  experimental  verification  by  Miller.  It  appears  from  this  theory  that  the  maximum 
of  intensity  is  less  sharply  marked  than  the  ordinary  theory  would  indicate,  and  does  not 
correspond  to  the  geometrical  minimum  of  deviation,  but  to  a  deviation  sensibly  greater. 
Also  that  the  region  of  sensible  illumination  extends  beyond  this  geometrical  minimum 
and  shades  off  gradually. 


RAINBOW. 


1085 


Fig.  766.— Production  of  Secondary  Bow. 


The  condensation  which  accompanies  minimum  deviation,  is  merely 
a  particular  case  of  the  general  mathematical  law  that  magnitudes 
remain  nearly  constant  in  the 
neighbourhood  of  a  maximum 
or  minimum  value.  The  rays 
which  compose  a  small  parallel 
pencil  S I  incident  at  and 
around  the  precise  point  which 
corresponds  to  minimum  devia- 
tion, will  thus  have  deviations 
which  may  be  regarded  as 
equal,  and  will  accordingly  re- 
main sensibly  parallel  at  emer- 
gence. A  parallel  pencil  inci- 
dent on  any  other  part  of  the  drop,  will  be  divergent  at  emergence. 

The  indices  of  refraction  for  red  and  violet  rays  from  air  into 
water  are  respectively 
-Vf  and  -W_,  and  calcu- 
lation shows  that  the 
distances  from  the  cen- 
tre of  the  sun  to  the 
parts  of  the  bow  in 
which  these  colours  are 
strongest  should  be  the 
supplements  of  42°  2' 
and  40°  17'  respectively. 
These  results  agree  with 
observation.  The  angles 
42°  2' and  40°  17' are  the 
distances  from  the  anti- 
solar  point,  which  is 
always  the  centre  of  the 
bow. 

The  rays  which  form 
the  secondary  bow  have 

undergone  two  internal  reflections,  as  represented  in  Fig.  7GC,  and 
here  again  a  special  concentration  occurs  in  the  direction  of  mini- 
mum deviation.  This  deviation  is  greater  than  180°  and  is  greatest 
for  the  most  refrangible  rays.  The  distance  of  the  arc  thus  formed 
from  the  sun's  centre,  is  3GO°  minus  the  deviation,  and  is  accord- 


Fig.  767. -Relative  Positions. 


1086  DISPERSION.      STUDY   OF   SPECTRA. 

ingly  least  for  the  most  refrangible  rays.  Thus  the  violet  arc 
is  nearest  the  sun,  and  the  red  furthest  from  it,  in  the  secondary 
bow. 

Some  idea  of  the  relative  situations  of  the  eye,  the  sun,  and  the 
drops  of  water  in  which  the  two  bows  are  formed,  may  be  obtained 
from  an  inspection  of  Fig.  767. 

SUNDRY  ADDITIONS  TO  PREVIOUS   CHAPTERS. 

1066A.  Goniometers. — A  goniometer  is  an  instrument  for  measuring 
the  angle  between  two  plane  faces  either  of  a  crystal  or  of  a  prism. 
The  measurement  is  usually  made  by  means  of  reflections  from  the 
two  faces.  This  may  be  done  in  either  of  the  two  following  ways. 
For  convenience  of  description  we  shall  assume  that  the  edge  in 
which  the  two  faces  meet  is  vertical;  in  practice  it  may  have  any 
direction. 

First  method. — Observe  in  one  of  the  two  faces  the  reflection  of 
an  object  at  a  few  yards'  distance,  in  the  same  horizontal  plane 
with  the  prism;  and  by  rotating  the  prism  in  this  plane  bring  the 
image  into  apparent  coincidence  with  some  other  object;  or,  if 
preferred,  bring  it  upon  the  cross-wires  of  a  fixed  telescope.  Then 
rotate  the  prism  in  the  horizonal  plane  till  the  other  face  gives  an 
image  of  the  same  object  in  the  same  position.  The  second  face  is 
now  in  or  parallel  to  the  position  previously  occupied  by  the  first 
face,  and  the  angle  through  which  the  prism  has  been  turned  is  the 
angle  between  one  face  and  the  other  face  produced.  By  subtracting 
it  from  180°  we  obtain  the  required  angle  between  the  faces.  The 
goniometer  is  furnished  with  a  graduated  circle  on  which  the  rotation 
is  read  off. 

Second  method. — The  goniometer  must  have  a  telescope  (with 
cross- wires)  which  can  travel  round  the  graduated 
circle,  while   always  directed   towards   its  centre, 
where  the  prism  stands.     The  prism  is  placed  in 
such  a  position  that  rays  from  a  distant  object,  or 
more  conveniently  from  a  slit  in  the  focus  of  a 
Fig.  767A.— Measure-    collimating  lens,  fall  upon  both  faces  simultaneously. 
prei"m0f  Angle  of    The  telescope  is  first  placed  so  as  to  receive  on  its 
cross-wires  the  image  formed  by  reflection  at  one 
face,  and  is  then  moved  past  the  base  of  the  prism  till  it  receives  the 
image  formed  by  reflection  at  the  other  face.     The  angle  through 


CONVEX  AND  CONCAVE  MIRRORS.  1087* 

which  it  has  been  moved  is  double  the  angle  of  the  prism,  as  is 
obvious  from  Fig.  767A,  in  which  the  directions  of  the  incident  and 
reflected  rays  are  represented  by  lines  marked  with  arrowheads. 
The  incident  rays  being  parallel,  a+/3  is  the  angle  of  the  prism,  and 
2a  +  2/3  is  the  angle  between  the  reflected  rays. 

After  measuring  the  angle  of  the  prism  by  means  of  the  observing 
telescope  and  collimator  with  slit,  the  index  of  refraction  of  the 
prism  can  be  determined  by  illuminating  the  slit  with  sodium  light 
or  some  other  monochromatic  light,  and  observing  with  the  telescope 
the  minimum  deviation  of  the  image  of  the  slit  formed  by  refraction 
through  the  prism.  The  index  of  refraction  for  the  particular  light 
employed  can  then  be  deduced  by  formula  (4)  page  1007. 

1066B.  Relation  between  Convex  and  Concave  Mirrors. — The  student 
should  notice  that  every  diagram  relating  to  reflection  from  a  concave 
mirror  is  equally  applicable  to  a  convex  mirror.  For  example  Fig.  683, 
page  988,  correctly  represents  the  virtual  image  AB  of  a  real  object 
ba  in  front  of  a  convex  mirror  having  C  for  its  centre  of  curvature 
and  F  for  its  principal  focus;  and  Fig.  677,  page  982,  shows  the  effect 
of  interposing  a  convex  mirror  in  the  path  of  rays  which  are  on  their 
way  to  form  a  real  image  ab.  The  effect  is  to  produce  the  virtual 
image  AB.  Or  we  may  take  AB  as  representing  the  real  image 
which  the  rays  were  on  their  way  to  form,  and  then  a  b  will  be  the 
virtual  image  which  is  formed  instead.  If  the  real  image  falls 
between  the  principal  focus  F  and  the  convex  mirror,  the  effect  will 
be  that  instead  of  an  inverted  virtual  image,  an  erect  real  image  will 
be  formed.  Thus  in  Fig.  683,  if  AB  be  the  image  which  the  rays 
were  on  their  way  to  form,  the  real  image  ba  will  be  formed  instead. 
Every  diagram  of  an  object  and  its  image,  as  formed  by  a  spherical 
mirror,  has  in  fact  four  different  interpretations,  since  the  object  and 
image  may  be  interchanged,  and  the  spherical  surface  may  be  polished 
on  either  side. 

1066C.  Nodal  Points  of  a  lens  or  system  of  lenses.  When  the  thick- 
ness of  a  lens  is  considerable  in  comparison  with  the  radii  of  curvature 
of  its  faces,  the  approximate  assumption  made  in  the  first  paragraph 
of  page  1016,  "that  rays  which  pass  through  the  centre  of  a  lens 
undergo  no  deviation"  is  no  longer  admissible. 

Referring  to  Fig.  718,  page  1015,  if  we  suppose  the  incident  and 
emergent  rays  SI  and  RE  to  be  produced  to  meet  the  axis  in  points 
N1  and  N2,  the  ultimate  positions  of  Nj  N2  when  the  rays  make  very 
small  angles  with  the  axis  are  called  the  ncdal  points  of  the  lens. 


1088*  NODAL  POINTS. 

They  are  the  two  images  of  the  centre  of  the  lens  formed  by  refraction 
out  of  the  lens  into  air  at  the  two  surfaces.  Whenever  the  incident 
ray  passes  through  the  first  nodal  point,  the  emergent  ray  passes 
through  the  second  nodal  point  and  is  parallel  to  the  incident  ray. 
Every  system  of  lenses  having  a  common  axis,  whether  the  lenses 
are  in  contact  with  each  other  or  at  any  distances  apart,  has  two 
nodal  points  possessing  the  above  property.  An  obvious  deduction 
from  this  property  is,  that  the  image  subtends  the  same  angle  at 
the  second  nodal  point  as  the  object  subtends  at  the  first.  The 
second  nodal  point  of  the  normal  human  eye  is  in  the  crystalline  lens 
near  the  back,  and  the  image  of  a  distant  object  formed  on  the  retina 
subtends  the  same  angle  at  this  point  which  the  object  subtends  at 
the  eye.  The  "  line  of  collimation  "  of  a  telescope,  which  we  have 
defined  on  page  1058  as  "  the  line  joining  the  cross  to  the  optical 
centre  of  the  object-glass,"  would  be  still  more  accurately  defined  as 
the  line  joining  the  cross  to  the  second  nodal  point  of  the  object- 
glass. 


CHAPTEK    LXXIII. 


COLOUR. 


1067.  Colour  as  a  Property  of  Opaque  Bodies. — A  body  which  reflects 
(by  irregular  reflection)  all  the  rays  of  the  spectrum  in  equal  propor- 
tion, will  appear  of  the  same  colour  as  the  light  which  falls  upon  it; 
that  is  to  say,  in  ordinary  cases,  white  or  gray.   But  the  majority  of 
bodies  reflect  some  rays  in  larger  proportion  than  others,  and  are 
therefore  coloured,  their  colour  being  that  which  arises  from  the 
mixture  of  the  rays  which  they  reflect.     A  body  reflecting  no  light 
would  be  perfectly  black.     Practically,  white,  gray,  and  black  differ 
only  in  brightness.    A  piece  of  white  paper  in  shadow  appears  gray, 
and  in  stronger  shadow  black. 

1068.  Colour  of  Transparent  Bodies. — A  transparent  body,  seen  by 
transmitted  light,  is  coloured,  if  it  is  more  transparent  to  some  rays 
than  to  others,  its  colour  being  that  which  results  from  mixing  the 
transmitted  rays.     No  new  ingredient  is  added  by  transmission,  but 
certain  ingredients  are  more  or  less  completely  stopped  out. 

Some  transparent  substances  appear  of  very  different  colours 
according  to  their  thickness.  A  solution  of  chloride  of  chromium, 
for  example,  appears  green  when  a  thin  layer  of  it  is  examined,  while 
a  greater  thickness  of  it  presents  the  appearance  of  reddish  brown. 
In  such  cases,  different  kinds  of  rays  successively  disappear  by  selec- 
tive absorption,  and  the  transmitted  light,  being  always  the  sum  of 
the  rays  which  remain  unabsorbed,  is  accordingly  of  different  com- 
position according  to  the  thickness. 

When  two  pieces  of  coloured  glass  are  placed  one  behind  the  other, 
the  light  which  passes  through  both  has  undergone  a  double  process 
of  selective  absorption,  and  therefore  consists  mainly  of  those  rays 
which  are  abundantly  transmitted  by  both  glasses;  or  to  speak 
broadly,  the  colour  which  we  see  in  looking  through  the  combination 


1088  COLOUR. 

is  not  the  sum  of  the  colours  of  the  two  glasses,  but  their  common 
part.  Accordingly,  if  we  combine  a  piece  of  ordinary  red  glass, 
transmitting  light  which  consists  almost  entirely  of  red  rays,  with  a 
piece  of  ordinary  green  glass,  which  transmits  hardly  any  red,  the 
combination  will  be  almost  black.  The  light  transmitted  through 
two  glasses  of  different  colour  and  of  the  same  depth  of  tint,  is 
always  less  than  would  be  transmitted  by  a  double  thickness  of 
either;  and  the  colour  of  the  transmitted  light  is  in  most  cases  a 
colour  which  occupies  in  the  spectrum  an  intermediate  place  between 
the  two  given  colours.  Thus,  if  the  two  glasses  are  yellow  and  blue, 
the  transmitted  light  will,  in  most  cases,  be  green,  since  most  natural 
yellows  and  blues  when  analysed  by  a  prism  show  a  large  quantity 
of  green  in  their  composition.  Similar  effects  are  obtained  by 
mixing  coloured  liquids. 

1069.  Colours  of  Mixed  Powders. — "In  a  coloured  powder,  each  par- 
ticle is  to  be  regarded  as  a  small  transparent  body  which  colours  light 
by  selective  absorption.  It  is  true  that  powdered  pigments  when 
taken  in  bulk  are  extremely  opaque.  Nevertheless,  whenever  we 
have  the  opportunity  of  seeing  these  substances  in  compact  and 
homogeneous  pieces  before  they  have  been  reduced  to  powder,  we 
find  them  transparent,  at  least  when  in  thin  slices.  Cinnabar, 
chromate  of  lead,  verdigris,  and  cobalt  glass  are  examples  in  point. 

"When  light  falls  on  a  powder  thus  composed  of  transparent  par- 
ticles, a  small  part  is  reflected  at  the  upper  surface;  the  rest  penetrates, 
and  undergoes  partial  reflection  at  some  of  the  surfaces  of  separation 
between  the  particles.  A  single  plate  of  uncoloured  glass  reflects  •£-% 
of  normally  incident  light;  two  plates  TV,  and  a  large  number  nearly 
the  whole.  In  the  powder  of  such  glass,  we  must  accordingly  con- 
clude that  only  about  -^  of  normally  incident  light  is  reflected  from 
the  first  surface,  and  that  all  the  rest  of  the  light  which  gives  the 
powder  its  whiteness  comes  from  deeper  layers.  It  must  be  the 
same  with  the  light  reflected  from  blue  glass;  and  in  coloured 
powders  generally  only  a  very  small  part  of  the  light  which  they 
reflect  comes  from  the  first  surface;  it  nearly  all  comes  from  beneath. 
The  light  reflected  from  the  first  surface  is  white,  except  when  the 
reflection  is  metallic.  That  which  comes  from  below  is  coloured,  and 
so  much  the  more  deeply  the  further  it  has  penetrated.  This  is  the 
reason  why  coarse  powder  of  a  given  material  is  more  deeply  col- 
oured than  fine,  for  the  quantity  of  light  returned  at  each  successive 
reflection  depends  only  on  the  number  of  reflections  and  not  on  the 


MIXTURE  OF  COLOURS.  1089 

thickness  of  the  particles.  If  these  are  large,  the  light  must  penetrate 
so  much  the  deeper  in  order  to  undergo  a  given  number  of  reflections, 
and  will  therefore  be  the  more  deeply  coloured. 

"The  reflection  at  the  surfaces  of  the  particles  is  weakened  if  we 
interpose  between  them,  in  the  place  of  air,  a  fluid  whose  index  of 
refraction  more  nearly  approaches  their  own.  Thus  powders  and 
pigments  are  usually  rendered  darker  by  wetting  them  with  water, 
and  still  more  with  the  more  highly  refracting  liquid,  oil. 

"If  the  colours  of  powders  depended  only  on  light  reflected  from 
their  first  surfaces,  the  light  reflected  from  a  mixed  powder  would 
be  the  sum  of  the  lights  reflected  from  the  surfaces  of  both.  But 
most  of  the  light,  in  fact,  comes  from  deeper  layers,  and  having  had 
to  traverse  particles  of  both  powders,  must  consist  of  those  rays 
which  are  able  to  traverse  both.  The  resultant  colour  therefore,  as 
in  the  case  of  superposed  glass  plates,  depends  not  on  addition  but 
rather  on  subtraction.  Hence  it  is  that  a  mixture  of  two  pigments 
is  usually  much  more  sombre  than  the  pigments  themselves,  if  these 
are  very  unlike  in  the  average  refrangibility  of  the  light  which  they 
reflect.  Vermilion  and  ultramarine,  for  example,  give  a  black-gray 
(showing  scarcely  a  trace  of  purple,  which  would  be  the  colour 
obtained  by  a  true  mixture  of  lights),  each  of  these  pigments  being 
in  fact  nearly  opaque  to  the  light  of  the  other." 1 

1070.  Mixtures  of  Colours. — By  the  colour  resulting  from  the  mix- 
ture of  two  lights,  we  mean  the  colour  which  is  seen  when  they  both 
fall  on  the  same  part  of  the  retina.  Propositions  regarding  mixtures 
of  colours  are  merely  subjective.  The  only  objective  differences  of 
colour  are  differences  of  refrangibility,  or  if  traced  to  their  source, 
differences  of  wave-frequency.  All  the  colours  in  a  pure  spectrum 
are  objectively  simple,  each  having  its  own  definite  period  of  vibra- 
tion by  which  it  is  distinguished  from  all  others.  But  whereas,  in 
acoustics,  the  quality  of  a  sound  as  it  affects  the  ear  varies  with 
every  change  in  its  composition,  in  colour,  on  the  other  hand,  very 
different  compositions  may  produce  precisely  the  same  visual  im- 
pression. Every  colour  that  we  see  in  nature  can  be  exactly  imitated 
by  an  infinite  variety  of  different  combinations  of  elementary  rays. 

To  take,  for  example,  the  case  of  white.  Ordinary  white  light 
consists  of  all  the  colours  of  the  spectrum  combined;  but  any  one  of 
the  elementary  colours,  from  the  extreme  red  to  a  certain  point  in 
yellowish  green,  can  be  combined  with  another  elementary  colour 

1  Translated  from  Helmholtz's  Pkytidogical  Optict,  §  20. 


1090 


COLOUR. 


on  the  other  side  of  green  in  such  proportion  as  to  yield  a  perfect 
imitation  of  ordinary  white.  The  prism  would  instantly  reveal  the 
differences,  but  to  the  naked  eye  all  these  whites  are  completely 
undistinguishable  one  from  another. 

1071.  Methods  of  Mixing  Colours. — The  following  are  some  of  the 
best  methods  of  mixing  colours  (that  is  coloured  lights): — 

1.  By  combining  reflected  and  transmitted  light;  for  example,  by 
looking  at  one  colour  through  a  piece  of  glass,  while  another  colour 
is  seen  by  reflection  from  the  near  side  of  the  glass.  The  lower 
sash  of  a  window,  when  opened  far  enough  to  allow  an  arm  to  be 
put  through,  answers  well  for  this  purpose.  The  brighter  of  the  two 
coloured  objects  employed  should  be  held  inside  the  window,  and 
seen  by  reflection;  the  second  object  should  then  be  held  outside  in 
such  a  position  as  to  be  seen  in  coincidence  with  the  image  of  the 
first.  As  the  quantity  of  reflected  light  increases  with  the  angle  of 
incidence,  the  two  colours  may  be  mixed  in  various  proportions 
by  shifting  the  position  of  the  eye.  This  method  is  not  however 

adapted  to  quantitative 
comparison,  and  can 
scarcely  be  employed 
for  combining  more  than 
two  colours. 

2.  By  employing  a  ro- 
tating disc  (Fig.  768) 
composed  of  differently 
coloured  sectors.  If  the 
disc  be  made  to  revolve 
rapidly,  the  sectors  will 
not  be  separately  visible, 
but  their  colours  will 
appear  blended  into  one 
on  account  of  the  per- 
sistence of  visual  impres- 
Fig.  70s. -Eotating  Disc.  sions.  The  proportions 

can  be  varied  by  varying 

the  sizes  of  the  sectors.  Coloured  discs  of  paper,  each  having  a 
radial  slit,  are  very  convenient  for  this  purpose,  as  any  moderate 
number  of  such  discs  can  be  combined,  and  the  sizes  of  the  sectors 
exhibited  can  be  varied  at  pleasure. 

The  mixed  colour  obtained  by  a  rotating  disc  is  to  be  regarded  as 


MIXTURE  OF  COLOURS.  1091 

a  mean  of  the  colours  of  the  several  sectors  —  a  mean  in  which  each 
of  these  colours  is  assigned  a  weight  proportional  to  the  size  of  its 
sector.  Thus,  if  the  360  degrees  which  compose  the  entire  disc 
consist  of  100°  of  red  paper,  100°  of  green,  and  160°  of  blue,  the 
intensity  of  the  light  received  from  the  red  when  the  disc  is  rotating 
will  be  only  $%•  of  that  which  would  be  received  from  the  red  sector 
when  seen  at  rest;  and  the  total  effect  on  the  retina  is  represented  by 
-$%  of  the  intensity  of  the  red,  plus  -J-£  of  the  intensity  of  the  green, 
plus  ^$  of  the  intensity  of  the  blue;  or  if  we  denote  the  colours  of 
the  sectors  by  their  initial  letters,  the  effect  may  be  symbolized  by 

the  formula  10R+1g6G+  16B.  Denoting  the  resultant  colour  by  C,  we 
have  the  symbolic  equation 


and  the  resultant  colour  may  be  called  the  mean  of  10  parts  of  red, 
10  of  green,  and  16  of  blue.  Colour-equations,  such  as  the  above, 
are  frequently  employed,  and  may  be  combined  by  the  same  rules 
as  ordinary  equations. 

3.  By  causing  two  or  more  spectra  to  overlap.  We  thus  obtain 
mixtures  which  are  the  sums  of  the  overlapping  colours. 

If,  in  the  experiment  of  §  1048,  we  employ,  instead  of  a  single 
straight  slit,  a  pair  of  slits  meeting  at  an  angle,  so  as  to  form  either 
an  X  or  a  V,  we  shall  obtain  mixtures  of  all  the  simple  colours  two 
and  two,  since  the  coloured  images  of  one  of  the  slits  will  cross  those 
of  the  other.  The  display  of  colours  thus  obtained  upon  a  screen  is 
exquisitely  beautiful,  and  if  the  eye  is  placed  at  any  point  of  the 
image  (for  example,  by  looking  through  a  hole  in  the  screen),  the 
prism  will  be  seen  rilled  with  the  colour  which  falls  on  this  point. 

1072.  Experiments  of  Helmholtz  and  Maxwell.  —  Helmholtz,  in  an 
excellent  series  of  observations  of  mixtures  of  simple  colours,  em- 
ployed a  spectroscope  with  a  V-shaped  slit,  the  two  strokes  of  the 
V  being  at  right  angles  to  one  another;  and  by  rotating  the  V  he 
was  able  to  diminish  the  breadth  and  increase  the  intensity  of  one 
of  the  two  spectra,  while  producing  an  inverse  change  in  the  other. 
To  isolate  any  part  of  the  compound  image  formed  by  the  two  over- 
lapping spectra,  he  drew  his  eye  back  from  the  eye-piece,  so  as  to 
limit  his  view  to  a  small  portion  of  the  field. 

But  the  most  effective  apparatus  for  observing  mixtures  of  simple 
colours  is  one  devised  by  Professor  Clerk  Maxwell,  by  means  of 
which  any  two  or  three  colours  of  the  spectrum  can  be  combined  in 


1092  COLOUR. 

any  required  proportions.  In  principle,  this  method  is  nearly  equi- 
valent to  looking  through  the  hole  in  the  screen  in  the  experiment 
above  described. 

Let  P  (Fig.  769)  be  a  prism,  in  the  position  of  minimum  deviation; 

L  a  lens;  E  and  R  conju- 
gate foci  for  rays  of  a 
particular  ref  rangibility* 
say  red;  E  and  V  conju- 
gate foci  for  rays  of  an- 
other given  refrangibi- 

Fig.  7G9. -Principle  of  Maxwell's  Colour-box.  lity,  Say  violet.      If  a  slit 

is  opened  at  R,  an  eye 

at  E  will  receive  only  red  rays,  and  will  see  the  lens  filled  with  red 
light.  If  this  slit  be  closed,  and  a  slit  opened  at  V,  the  eye,  still 
placed  at  E,  will  see  the  lens  filled  with  violet  light.  If  both  slits 
be  opened,  it  will  see  the  lens  filled  with  a  uniform  mixture  of  the 
two  lights;  and  if  a  third  slit  be  opened,  between  R  and  V,  the  lens 
will  be  seen  filled  with  a  mixture  of  three  lights. 

Again,  from  the  properties  of  conjugate  foci,  if  a  slit  is  opened  at 
E,  its  spectral  image  will  be  formed  at  R  V,  the  red  part  of  it  being 
at  R,  and  the  violet  part  at  V. 

The  apparatus  was  inclosed  in  a  box  painted  black  within.  There 
was  a  slit  fixed  in  position  at  E,  and  a  frame  with  three  movable 
slits  at  RV.  When  it  was  desired  to  combine  colours  from  three 
given  parts  of  the  spectrum,  specified  by  reference  to  Fraunhofer's 
lines,  the  slit  E  was  first  turned  towards  the  light,  giving  a  real 
spectrum  in  the  plane  R  V,  in  which  Fraunhofer's  lines  were  visible, 
and  the  three  movable  slits  were  set  at  the  three  specified  parts  of 
the  spectrum.  The  box  was  then  turned  end  for  end,  so  that  light 
was  admitted  (reflected  from  a  large  white  screen  placed  in  sunshine) 
at  the  movable  slits,  and  the  observer,  looking  in  at  the  slit  E,  saw 
the  resultant  colour. 

1073.  Results  of  Experiment. — The  following  are  some  of  the  prin- 
cipal results  of  experiments  on  the  mixture  of  coloured  lights: — 

1.  Lights  which  appear  precisely  alike  to  the  naked  eye  yield 
identical  results  in  mixtures;  or  employing  the  term  similar  to 
express  apparent  identity  as  judged  by  the  naked  eye,  the  sums  of 
similar  lights  are  themselves  similar.  It  is  by  reason  of  this  phy- 
sical fact,  that  colour-equations  yield  true  results  when  combined 
according  to  the  ordinary  rules  of  elimination. 


MIXTURE  OP  COLOURS.  1093 

In  the  strict  application  of  this  rule,  the  same  observer  must  be 
the  judge  of  similarity  in  the  different  cases  considered.     For 

2.  Colours  may  be  similar  as  seen  by  one  observer,  and  dissimilar 
as  seen  by  another;  and  in  like  manner,  colours  may  be  similar  as 
seen  through  one  coloured  glass,  and  dissimilar  as  'seen  through 
another.      The   reason,  in  both  cases,  is  that   selective  absorption 
depends  upon  real  composition,  which  may  be  very  different  for  two 
merely  similar  lights.      Most  eyes  are  found  to  exhibit  selective 
absorption  of  a  certain  kind  of  elementary  blue,  which  is  accord- 
ingly weakened  before  reaching  the  retina. 

3.  Between   any   four   colours,   given    in   intensity   as   well   as 
in   kind,  one   colour-equation  subsists;   expressing   the   fact   that, 
when  we  have  the  power  of  varying  their  intensities  at  pleasure, 
there  is  one  definite  way  of  making  them  yield  a  'match,  that  is 
to  say,  a  pair  of   similar   colours.      Any  colour  can  therefore  be 
completely  specified   by  three  numbers,  expressing  its  relation  to 
three  arbitrarily  selected  colours.     This  is  analogous  to  the  theorem 
in  statics  that  a  force  acting  at  a  given  point  can  be  specified  by 
three  numbers  denoting  its  components  in  three  arbitrarily  selected 
directions. 

4.  Between  any  five  colours,  given  in  intensity  as  well  as  in  kind, 
a  match  can  be  made  in  one  definite  way  by  taking  means;1  for 
example,  by  mounting  the  colours  on  two  rotating  discs.    If  we  had 
the  power  of  illuminating  one  disc  more  strongly  than  the  other  in 
any  required  ratio,  four  colours  would  be  theoretically  sufficient;  and 
we  can,  in  fact,  do  what  is  nearly  equivalent  to  this,  by  employing 
black  as  one  of  our  five  colours.     Taking  means  of  colours  is  analo- 
gous to  finding  centres  of  gravity.     In  following  out  the  analogy,  a 
colour  given  in  kind  merely  must   be  represented  by  a  material 
point  given  in  position  merely,  and  the  intensity  of  the  colour  must 
be  represented  by  the  mass  of  the  material  point.     The  means  of 
two  given  colours  will  be  represented  by  points  in  the  line  joining 
two  given  points.     The  means  of  three  given  colours  will  be  repre- 
sented by  points  lying  within  the  triangle  formed  by  joining  three 
given  points,  and  the  means  of  four  given  colours  will  be  repre- 
sented by  points  within  a  tetrahedron  whose  four  corners  are  given. 
When  we  have  five  colours  given,  we  have  five  points  given,  and  of 
these  generally  no  four  will  lie  in  one  plane.    Call  them  A,  B,  C,  D,  E. 

1  Propositions  4  and  5  are  not  really  independent,  but  represent  different  aspects  of  one 
physical  {or  rather  physiological)  law. 


1094  COLOUR. 

Then  if  E  lies  within  the  tetrahedron  A  B  C  D,  we  can  make  the 
centre  of  gravity  of  A,  B,  C,  and  D  coincide  with  E,  and  the  colour 
E  can  be  matched  by  a  mean  of  the  other  four  colours. 

If  E  lies  outside  the  tetrahedron,  it  must  be  situated  at  a  point 
from  which  either  one,  two,  or  three  faces  are  visible  (the  tetrahedron 
being  regarded  as  opaque). 

If  only  one  face  is  visible,  let  it  be  B  C  D;  then  the  point  where 
the  straight  line  E  A  cuts  BCD  is  the  match;  for  it  is  a  mean  of 
E  and  A,  and  is  also  a  mean  of  B,  C,  and  D. 

If  two  faces  are  visible,  let  them  be  A  C  D  and  BCD;  then 
the  intersection  of  the  edge  C  D  with  the  plane  E  A  B  is  the 
match. 

If  three  faces  are  visible,  let  them  be  the  three  which  meet  at  A; 
then  A  is  the  match,  for  it  lies  within  the  tetrahedron  E  B  C  D. 

With  six  given  colours,  combined  five  at  a  time,  six  different 
matches  can  be  made,  and  six  colour-equations  will  thus  be  obtained, 
the  consistency  of  which  among  themselves  will  be  a  test  of  the 
accuracy  both  of  theory  and  observation,  as  only  three  of  the  six  can 
be  really  independent.  Experiments  which  have  been  conducted  on 
this  plan  have  given  very  consistent  results. 

1074.  Cone  of  Colour. — All  the  results  of  mixing  colours  can 
be  represented  geometrically  by  means  of  a  cone  or  pyramid 
within  which  all  possible  colours  will  have  their  definite  places. 
The  vertex  will  represent  total  blackness,  or  the  complete  absence 
of  light;  and  colours  situated  on  the  same  line  passing  through 
the  vertex  will  differ  only  in  intensity  of  light.  Any  cross- 
section  of  the  cone  will  contain  all  colours,  except  so  far  as  intensity 
is  concerned,  and  the  colours  residing  on  its  perimeter  will  be  the 
colours  of  the  spectrum  ranged  in  order,  with  purple  to  fill  up  the 
interval  between  violet  and  red.  It  appears  from  Maxwell's  experi- 
ments, that  the  true  form  of  the  cross-section  is  approximately 
triangular;1  with  red,  green,  and  violet  at  the  three  corners.  When 
all  the  colours  have  been  assigned  their  proper  places  in  the  cone, 
a  straight  line  joining  any  two  of  them  passes  through  colours 
which  are  means  of  these  two;  and  if  two  lines  are  drawn  from  the 
vertex  to  any  two  colours,  the  parallelogram  constructed  on  these 
two  lines  will  have  at  its  further  corner  the  colour  which  is  the  sum 
of  these  two  colours.  A  certain  axial  line  of  the  cone  will  contain 

1  The  shape  of  the  triangle  is  a  mere  matter  of  convenience,  not  involving  any  question 
of  fact 


COMPLEMENTARY  COLOURS.  1095 

white  or  gray  at  all  points  of  its  length,  and  may  be  called  the  line 
of  white. 

It  is  convenient  to  distinguish  three  qualities  of  colour  which  may 
be  called  hue,  depth,  and  brightness.  Brightness  or  intensity  of  light 
is  represented  by  distance  from  the  vertex  of  the  cone.  Depth 
depends  upon  angular  distance  from  the  line  of  white,  and  is  the 
same  for  all  points  on  the  same  line  through  the  vertex.  Paleness 
or  lightness  is  the  opposite  of  depth,  and  is  measured  by  angular 
nearness  to  the  line  of  white.  Hue  or  tint  is  that  which  is  often 
par  excellence  termed  colour.  If  we  suppose  a  plane,  containing  the 
line  of  white,  to  revolve  about  this  line  as  axis,  it  will  pass  succes- 
sively through  different  tints;  and  in  any  one  position  it  contains 
only  two  tints,  which  are  separated  from  each  other  by  the  line  of 
white,  and  are  complementary. 

Red  is  complementary  to Bluish  green. 

Orange         „  „         Sky  blue. 

Yellow         „  „         Violet  blue. 

Greenish  yellow        „        Violet. 

Green  , Pink. 

Any  two  colours  of  complementary  tint  give  white,  when  mixed  in 
proper  proportions;  and  any  three  colours  can  be  mixed  in  such 
proportions  as  to  yield  white,  if  the  triangle  formed  by  joining  them 
is  pierced  by  the  line  of  white. 

Every  colour  in  nature,  except  purple,  is  similar  to  a  colour  of  the 
spectrum  either  pure  or  diluted  with  gray;  and  all  purples  are 
similar  to  mixtures  of  red  and  blue  with  or  without  dilution. 
Brown  can  be  imitated  by  diluting  orange  with  dark  gray.  The 
orange  and  yellow  of  the  spectrum  can  themselves  be  imitated  by 
adding  together  red  and  green. 

1075.  Three  Primary  Colour-sensations. — All  authorities  are  now 
agreed  in  accepting  the  doctrine,  first  propounded  by  Dr.  Thomas 
Young,  that  there  are  three  elements  of  colour-sensation;  or,  in  other 
words,  three  distinct  physiological  actions,  which,  by  their  various 
combinations,  produce  our  various  sensations  of  colour.  Each  is 
excitable  by  light  of  various  wave-lengths  lying  within  a  wide  range, 
but  has  a  maximum  of  excitability  for  a  particular  wave-length, 
and  is  affected  only  to  a  slight  degree  by  light  of  wave-length  very 
different  from  this.  The  cone  of  colour  is  theoretically  a  triangular 
pyramid,  having  for  its  three  edges  the  colours  which  correspond  to 
these  three  wave-lengths;  but  it  is  probable  that  we  cannot  obtain 


1096  COLOUR. 

one  of  the  three  elementary  colour-sensations  quite  free  from  admix- 
ture of  the  other  two,  and  the  edges  of  the  pyramid  are  thus  practi- 
cally rounded  off.  One  of  these  sensations  is  excited  in  its  greatest 
purity  by  the  green  near  Fraunhofer's  line  6,  another  by  the  extreme 
red,  and  the  third  by  the  extreme  violet. 

Helmholtz  ascribes  these  three  actions  to  three  distinct  sets  of 
nerves,  having  their  terminations  in  different  parts  of  the  thickness 
of  the  retina— a  supposition  which  aids  in  accounting  for  the  approxi- 
mate achromatism  of  the  eye,  for  the  three  sets  of  nerve-terminations 
may  thus  be  at  the  proper  distances  for  receiving  distinct  images  of 
red,  green,  and  violet  respectively,  the  focal  length  of  a  lens  being 
shorter  for  violet  than  for  red. 

Light  of  great  intensity,  whatever  its  composition,  seems  to  pro- 
duce a  considerable  excitement  of  all  three  elements  of  colour-sensa- 
tion. If  a  spectroscope,  for  example,  be  directed  first  to  the  clouds 
and  then  to  the  sun,  all  parts  of  the  spectrum  appear  much  paler  in 
the  latter  case  than  in  the  former. 

The  popular  idea  that  red,  yellow,  and  blue  are  the  three  prima- 
ries, is"  quite  wrong  as  regards  mixtures  of  lights  or  combinations  of 
colour-sensations.  The  idea  has  arisen  from  facts  Observed  in  con- 
nection with  the  mixture  of  pigments  and  the  transmission  of  light 
through  coloured  glasses.  We  have  already  pointed  out  the  true 
interpretation  of  observations  of  this  nature,  and  have  only  now  to 
add  that  in  attempting  to  construct  a  theory  of  the  colours  obtained 
by  mixtures  of  pigments,  the  law  of  substitution  of  similars  cannot 
be  employed.  Two  pigments  of  similar  colour  will  not  in  general 
give  the  same  result  in  mixtures. 

1076.  Accidental  Images. — If  we  look  steadily  at  a  bright  stained- 
glass  window,  and  then  turn  our  eyes  to  a  white  wall,  we  see  an 
image  of  the  window  with  the  colours  changed  into  their  com- 
plementaries.  The  explanation  is  that  the  nerves  which  have  been 
strongly  exercised  in  the  perception  of  the  bright  colours  have  had 
their  sensibility  diminished,  so  that  the  balance  of  action  which  is 
necessary  to  the  sensation  of  white  no  longer  exists,  but  those 
elements  of  sensation  which  have  not  been  weakened  preponderate. 
The  subjective  appearances  arising  from  this  cause  are  called  nega- 
tive accidental  images.  Many  well-known  effects  of  contrast  are 
similarly  explained.  White  paper,  when  seen  upon  a  background 
of  any  one  colour,  often  appears  tinged  with  the  complementary 
colour;  and  stray  beams  of  sunlight  entering  a  room  shaded  with 


COLOUR-BLINDNESS.  1097 

yellow  holland  blinds,  produce  blue  streaks  when  they  fall  upon  a 
white  tablecloth. 

In  some  cases,  especially  when  the  object  looked  at  is  painfully 
bright,  there  is  a  positive  accidental  image;  that  is,  one  of  the  same 
colour  as  the  object;  and  this  is  frequently  followed  by  a  negative 
image.  A  positive  accidental  image  may  be  regarded  as  an  extreme 
instance  of  the  persistence  of  impressions. 

1077.  Colour-blindness. — What  is  called  colour-blindness  has  been 
found,  in  every  case  which  has  been  carefully  investigated,  to  consist 
in  the  absence  of  the  elementary  sensation  corresponding  to  red. 
To  persons  thus  affected  the  solar  spectrum  appears  to  consist  of 
two  decidedly  distinct  colours,  with  white  or  gray  at  their  place  of 
junction,  which  is  a  little  way  on  the  less  refrangible  side  of  the 
line  F.     One  of  these  two  colours  is  doubtless  nearly  identical  with 
the  normal  sensation  of  blue  or  violet.     It  attains  its  maximum 
about  midway  between  F  and  G,  and  extends  beyond  G  as  far  as 
the  normally  visible  spectrum.    The  other  colour  extends  a  consider- 
able distance  into  what  to  normal  eyes  is  the  red  portion  of  the 
spectrum,  attaining  its  maximum  about  midway  between  D  and  E, 
and  becoming  deeper  and  more  faint  till  it  vanishes  at  about  the 
place  where  to  normal  eyes  crimson  begins.     The  scarlet  of  the 
spectrum  is  thus  visible  to  the  colour-blind,  not  as  scarlet  but  as  a 
deep  dark  colour,  perhaps  a  kind  of  dark  green,  orange  and  yellow 
as  brighter  shades  of  the  same  colour,  while  bluish-green  appears 
nearly  white. 

It  is  obvious  from  this  account  that  what  is  called  "  colour-blind- 
ness "  should  rather  be  called  dichroic  vision,  normal  vision  being 
distinctively  designated  as  trichroic.  To  the  dichroic  eye  any  colour 
can  be  matched  by  a  mixture  of  yellow  and  blue,  and  a  match  can 
be  made  between  any  three  (instead  of  four)  given  colours.  Objects 
which  have  the  same  colour  to  the  trichroic  eye  have  also  the  same 
colour  to  the  dichroic  eye. 

1078.  Colour  and  Musical  Pitch. — As  it  is  completely  established 
that  the  difference  between  the  colours  of  the  spectrum  is  a  differ- 
ence of  vibration-frequency,  there  is  an  obvious  analogy  between 
colour  and  musical  pitch;  but  in  almost  all  details  the  relations 
between  colours  are  strikingly  different  from  the  relations  between 
sounds. 

The  compass  of  visible  colour,  including  the  lavender  rays  which 
lie  beyond  the  violet,  and  are  perhaps  visible  not  in  themselves, 


1098  COLOUR. 

but  by  the  fluorescence  which  they  produce  on  the  retina,  is, 
according  to  Helmholtz,  about  an  octave  and  a  fourth;  but  if  we 
exclude  the  lavender,  it  is  almost  exactly  an  octave.  Attempts 
have  been  made  to  compare  the  successive  colours  of  the  spectrum 
with  the  notes  of  the  gamut;  but  much  forcing  is  necessary  to 
bring  out  any  trace  of  identity,  and  the  gradual  transitions  which 
characterize  the  spectrum,  and  constitute  a  feature  of  its  beauty,  are 
in  marked  contrast  to  the  transitions  per  saltum  which  are  required 
in  music. 


CHAPTER     LXXIV. 


WAVE  THEORY   OF  LIGHT. 


1079.  Principle  of  Huygens.1 — The  propagation  of  waves,  whether 
of  sound  or  light,  is  a  propagation  of  energy.  Each  small  portion  of 
the  medium  experiences  successive  changes  of  state,  involving  changes 
in  the  forces  which  it  exerts  upon  neighbouring  portions.  These 
,  changes  of  force  produce  changes  of  state  in  these  neighbouring  por- 
tions, or  in  such  of  them  as  lie  on  the  forward  side  of  the  wave,  and 
thus  a  disturbance  existing  at  any  one  part  is  propagated  onwards. 

Let  us  denote  by  the  name  wave-front  a  continuous  surface  drawn 
through  particles  which  have  the  same  phase;  then  each  wave-front 
advances  with  the  velocity  of  light,  and  each  of  its  points  may  be 
regarded  as  a  secondary  centre  from  which  disturbances  are  continu- 
ally propagated.  This  mode  of  regarding  the  propagation  of  light  is 
due  to  Huygens,  who  derived  from  it  the  following  principle,  which 
lies  at  the  root  of  all  practical  applications  of  the  undulatory  theory: 
The  disturbance  at  any  point  of  a  wave-front  is  the  resultant  (given 
by  the  parallelogram  of  motions)  of  the  separate  disturbances  which 
the  different  portions  of  the  same  wave- front  in  any  one  of  its 
earlier  positions,  would  have  occasioned  if  acting  singly.  This 
principle  involves  the  physical  fact  that  rays  of  light  are  not  affected 
by  crossing  one  another;  and  its  truth,  which  has  been  experiment- 
ally tested  by  a  variety  of  consequences,  must  be  taken  as  an  indica- 
tion that  the  amplitudes  of  luminiferous  vibrations  are  infinitesimal 
in  comparison  with  the  wave-lengths.  A  similar  law  applies  to  the 
resultant  of  small  disturbances  generally,  and  is  called  by  writers  on 
dynamics  the  law  of  "superposition  of  small  motions."  It  is  analo- 
gous to  the  arithmetical  principle  that,  when  a  and  b  are  very  small 
fractions,  the  product  of  1  +  a  and  1-1-6  may  be  identified  with 

1  For  the  spelling  of  this  name  see  remarks  by  Lalande,  Mimoirtt  de  I'AcacUmie,  1773. 


1100  WAVE   THEORY   OF  LIGHT. 

l+a+b;  the  term  a  6,  which  represents  the  mutual  influence  of  two 
small  changes,  being  negligible  in  comparison  with  the  sum  a  +  b  of 
the  small  changes  themselves. 

1080.  Explanation  of  Rectilinear  Propagation. —  In  a  medium  in 
which  light  travels  with  the  same  velocity  in  all  parts  and  in  all 
directions,  the  waves  propagated  from  any  point  will  be  concentric 
spheres,  having  this  point  for  centre;  and  the  lines  of  propagation, 
in  other  words  the  rays  of  light,  will  be  the  radii  of  these  spheres. 
It  can  in  fact  be  shown  that  the  only  part  of  one  of  these  waves 
which  needs  to  be  considered,  in  computing  the  resultant  disturbance 
of  an  external  point,  is  the  part  which  lies  directly  between  this 
external  point  and  the  centre  of  the  sphere.  The  remainder  of  the 
wave-front  can  be  divided  into  small  parts,  each  of  which,  by  the 
mutual  interference  of  its  own  subdivisions,  gives  a  resultant  effect 
of  zero  at  the  given  point.  We  express  these  properties  by  saying 
that  in  a  homogeneous  and  isotropic  medium  the  wave-surface  is  a 
sphere,  and  the  rays  are  normal  to  the  wave-fronts.  This  class  of 
media  includes  gases,  liquids,  crystals  of  the  cubic  system,  and  well- 
annealed  glass. 

If  a  medium  be  homogeneous  but  not  isotropic,  disturbances 
emanating  from  a  point  in  it  will  be  propagated  in  waves  which  will 
retain  their  form  unchanged  as  they  expand  in  receding  from  their 
source,  but  this  form  will  not  generally  be  spherical.  The  rays  of 
light  in  such  a  medium  will  be  straight,  proceeding  directly  from  the 
centre  of  disturbance,  and  any  one  ray  will  cut  all  the  wave-fronts 
at  the  same  angle;  but  this  angle  wrill  generally  be  different  for 
different  rays.  In  this  case,  as  in  the  last,  the  disturbance  produced 
at  any  point  may  be  computed  by  merely  taking  into  account  that 
small  portion  of  a  wave-front  which  lies  directly  between  the  given 
point  and  the  source, — in  other  wordst  which  lies  on  or  very  near  to 
the  ray  which  traverses  the  given  point. 

A  disturbance  in  such  a  medium  usually  gives  rise  to  two  sets  of 
waves,  having  two  distinct  forms,  and  these  remarks  apply  to  each 
set  separately. 

The  tendency  of  the  different  parts  of  a  wave-front  to  propagate 
disturbances  in  other  directions  besides  the  single  one  to  which  such 
propagation  is  usually  confined,  is  manifested  in  certain  phenomena 
which  are  included  under  the  general  name  of  diffraction. 

The  only  wave-fronts  with  which  it  is  necessary  to  concern  our- 
selves are  those  which  belong  to  waves  emanating  from  a  single 


HUYGENS    CONSTRUCTION. 


1101 


point, — that  is  to  say,  either  from  a  surface  really  very  small,  or 
from  a  surface  which,  by  reason  of  its  distance,  subtends  a  very 
small  solid  angle  at  the  parts  of  space  considered. 

1081.  Application  to  Refraction. — When  waves  are  propagated  from 
one  medium  into  another,  the  principle  of  Huygens  leads  to  the 
following  construction: — 

Let  A  E  (Fig.  770)  represent  a  portion  of  the  surface  of  separation 
between  two  media,  and  A  B  a  portion  of  a  wave-front  in  the  first 
medium;  both  portions  being  small  enough  to  be  regarded  as  plane. 


Fig.  770.—  Huygeus'  Construction  for  Wave-front. 

Then  straight  lines  C  A,  D  B  E,  normal  to  the  wave-front,  represent 
rays  incident  at  A  and  E.  From  A  as  centre,  describe  a  wave-surface, 
of  such  dimensions  that  light  emanating  from  A  would  reach  this 
surface  in  the  same  time  in  which  light  in  air  travels  the  distance 
B  E,  and  draw  a  tangent  plane  (perpendicular  to  the  plane  of  incid- 
ence) through  E  to  this  surface.  Let  F  be  the  point  of  contact 
(which  is  not  necessarily  in  the  plane  of  incidence).  Then  the  tan- 
gent plane  E  F  is  a  wave-front  in  the  second  medium,  and  A  F  is  a 
ray  in  the  second  medium;  for  it  can  be  shown  that  disturbances 
propagated  from  all  points  in  the  wave-front  AB  will  just  have 
reached  E  F  when  the  disturbance  propagated  from  B  has  reached  E. 
For  example,  a  ray  proceeding  from  m,  the  middle  point  of  the  line 
A  B,-  will  exhaust  half  the  time  in  travelling  to  the  middle  point  a 
of  A  E,  and  the  remaining  half  in  travelling  through  af,  equal  and 
parallel  to  half  of  AF. 

When  the  wave-surfaces  in  both  media  are  spherical,  the  planes  of 
incidence  and  refraction  ABE,  AFE  coincide,  the  angle  BAE 
(Fig.  771)  between  the  first  wave-front  and  the  surface  of  separation 
is  the  same  as  the  angle  between  the  normals  to  these  surfaces,  that 


1102 


WAVE  THEORY   OF  LIGHT. 


Fig.  771. — Wave-front  in  Ordinary  Refraction. 


is  to  say,  is  the  angle  of  incidence;  and  the  angle  AEF  between 
the  surface  of  separation  and  the  second  wave-front  is  the  angle  of 

refraction.     The  sine  of  the  former  is  ^-^,  and  the  sine  of  the  latter 

is  ^|.     The  ratio  ~,  is  therefore  J|.     But  B  E  and  A  F  are  the 

distances  travelled  in  the  same 
time  in  the  two  media.  Hence 
the  shies  of  the  angles  of  in- 
cidence and  refraction  are  di- 
rectly as  the  velocities  of  pro- 
pagation of  the  incident  and 
refracted  light.  The  relative 
index  of  refraction  from  one 
medium  into  another  is  there- 
fore the  ratio  of  the  velocity 
of  light  in  the  first  medium  to 
its  velocity  in  the  second;  and 

the  absolute  index  of  refraction  of  any  medium  is  inversely  as  the 

velocity  of  light  in  that  medium. 

1082.  Application  to  Reflection. — The  explanation  of  reflection  is 
precisely  similar.     Let  C  A,  D  E  (Fig.  772)  be  parallel  rays  incident 
at  A  and  E;  AB  the  wave- front.     As  the  successive  points  of  the 
wave-front  arrive  at   the  reflecting  surface,  hemispherical  waves 

diverge  from  the  points  of  in- 
cidence; and  by  the  time  that 
B  reaches  E,  the  wave  from  A 
will  have  diverged  in  all  direc- 
tions to  a  distance  equal  to 
B  E.  If  then  we  describe  in 
the  plane  of  incidence  a  semi- 
circle, with  centre  A  and  radius 
equal  to  B  E,  the  tangent  E  F 
to  this  semicircle  will  be  the 
wave -front  of  the  reflected 
light,  and  A  F  will  be  the  reflected  ray  corresponding  to  the  incident 
ray  CA.  From  the  equality  of  the  right-angled  triangles  ABE, 
E  F  A,  it  is  evident  that  the  angles  of  incidence  and  reflection  are 
equal. 

1083.  Newtonian  Explanation  of  Refraction. — In  the  Newtonian 
theory,  the  change  of  direction  which  a  ray  experiences  at  the  bound- 


rig.  772. -Wave-front  in  Reflection. 


FOUCAULT'S  CRUCIAL  EXPERIMENT.  1103 

ing  surface  of  two  media,  is  attributed  to  the  preponderance  of  the 
attraction  of  the  denser  medium  upon  the  particles  of-  light.  As  the 
resultant  force  of  this  attraction  is  normal  to  the  surface,  the  tan- 
gential component  of  velocity  remains  unchanged,  and  the  normal 
component  is  increased  or  diminished  according  as  the  incidence  is 
from  rare  to  dense  or  from  dense  to  rare.  Let  /i  denote  the  relative 
index  of  refraction  from  rare  to  dense.  Let  v,  v  be  the  velocities  of 
light  in  the  rarer  and  denser  medium  respectively,  and  i,  i'  the  angles 
which  the  rays  in  the  two  media  make  with  the  normal.  Then  the 
tangential  components  of  velocity  in  the  two  media  are  v  sin  i,  v  sin  i' 
respectively,  and  these  by  the  Newtonian  theory  are  equal;  whence 

v~sini'~'1'  whereas  according  to  the  undulatory  theory  -=\  In 
the  Newtonian  theory,  the  velocity  of  light  in  any  medium  is 
directly  as  the  absolute  index  of  refraction  of  the  medium;  whereas, 
in  the  undulatory  theory,  the  reverse  rule  holds. 

The  main  design  of  Foucault's  experiment  with  the  rotating  mirror 
(§  942),  in  its  original  form,  was  to  put  these  opposite  conclusions  to 
the  test  of  direct  experiment.  For  this  purpose  it  was  not  necessary 
to  determine  the  velocity  of  the  rotating  mirror,  since  it  affected 
both  the  observed  displacements  alike.  The  two  images  were  seen 
in  the  same  field  of  view,  and  were  easily  distinguished  by  the  green- 
ness of  the  water-image.  In  eveiy  trial  the  water-image  was  more 
displaced  than  the  air-image,  indicating  longer  time  and  slower  velo- 
city; and  the  measurements  taken  were  in  complete  accordance  with 
the  undulatory  theory,  while  the  Newtonian  theory  was  conclusively 
disproved. 

1084.  Principle  of  Least  Time. — The  path  by  which  light  travels 
from  one  point  to  another  is  in  the  generality  of  cases  that  which  occu- 
pies least  time.  For  example,  in  ordinary  cases  of  reflection  (except 
from  very  concave1  surfaces),  if  we  select  any  two  points,  one  on  the 
incident  and  the  other  on  the  reflected  ray,  the  sum  of  their  distances 
from  the  point  of  incidence  is  less  than  the  sum  of  their  distances 
from  any  neighbouring  point  on  the  reflecting  surface.  In  this  case, 
.since  only  one  medium  is  concerned,  distance  is  proportional  to  time. 
When  a  ray  in  air  is  refracted  into  water,  if  we  select  any  two  points, 

1  Suppose  an  ellipse  described,  having  the  two  selected  points  for  foci,  and  passing 
through  the  point  of  incidence.  If  the  curvature  of  the  reflecting  surface  in  the  plane  of 
incidence  is  greater  than  the  curvature  of  this  ellipse,  the  length  of  the  path  is  a  maximum, 
if  less,  a  minimum.  This  follows  at  once  from  the  constancy  of  the  sum  of  the  focal 
distances  in  an  ellipse. 
70 


1104  WAVE  THEORY  OF  LIGHT. 

one  on  the  incident  and  the  other  on  the  refracted  ray,  and  call  their 
distances  from  any  point  of  the  refracting  surface  s,  s'  respectively, 
and  the  velocities  of  propagation  in  the  two  media  v,  v,  then  the  sum 

of  *  and  *-,  is  generally  less  when  s  and  s'  are  measured  to  the  point 
of  incidence  than  when  they  are  measured  to  any  neighbouring  point 
on  the  surface.  ~v  is  evidently  the  time  of  going  from  the  first  point 
to  the  refracting  surface,  and  ^  the  time  from  the  refracting  surface 

to  the  second  point. 

The  proposition  as  above  enunciated  admits  of  certain  exceptions, 
the  time  being  sometimes  a  maximum  instead  of  a  minimum.  The 
really  essential  condition  (which  is  fulfilled  in  both  these  opposite 
cases)  is  that  all  points  on  a  small  area  surrounding  the  point  of 
incidence  give  sensibly  the  same  time.  The  component  waves  sent 
from  all  parts  of  this  small  area  will  be  in  the  same  phase,  and  will 
propagate  a  ray  of  light  by  their  combined  action. 

When  the  two  points  considered  are  conjugate  foci,  and  there  is 
no  aberration,  this  condition  must  be.  fulfilled  by  all  the  rays  which 
pass  through  both;  and  the  time  of  travelling  from  one  focus  to  the 
other  is  the  same  for  all  the  rays.  Spherical  waves  diverging  from 
one  focus  will,  after  incidence,  become  spherical  waves  converging  to 
or  diverging  from  its  conjugate  focus.  An  effect  of  this  kind  can  be 
beautifully  exhibited  to  the  eye  by  means  of  an  elliptic  dish  contain- 
ing mercury.  If  agitation  is  produced  at  one  focus  of  the  ellipse  by 
dipping  a  small  rod  into  the  liquid  at  this  point,  circular  waves  will 
be  seen  to  converge  towards  the  other  focus.  A  circular  dish  exhi- 
bits a  similar  result  somewhat  imperfectly;  waves  diverging  from  a 
point  near  the  centre  will  be  seen  to  converge  to  a  point  symmetri- 
cally situated  on  the  other  side  of  the  centre. 

When  the  second  point  lies  on  a  caustic  surface  formed  by  the 
reflection  or  refraction  of  rays  emanating  from  the  first  point,  all 
points  on  an  area  of  sensible  magnitude  in  the  neighbourhood  of  the 
point  of  incidence  would  give  sensibly  the  same  time  of  travelling 
as  the  actual  point  of  incidence,  so  that  the  light  which  traverses 
a  point  on  a  caustic  may  be  regarded  as  coming  from  an  area  of 
sensible  magnitude  instead  of  (as  in  the  case  of  points  not  on  the 
caustic)  an  excessively  small  area.  An  eye  placed  at  a  point  on  a 
caustic  will  see  this  portion  of  the  surface  filled  with  light. 

As  the  velocity  of  light  is  inversely  proportional  to  the  index  of 


ATMOSPHERIC   REFRACTION.  1105 

refraction  ft,  the  time  of  travelling  a  distance  8  with  constant  velocity 
may  be  represented  by  ps,  and  if  a  ray  of  light  passes  from  one  point 
to  another  by  a  crooked  path,  made  up  of  straight  lines  s1}82,83,  .  .  .  . 
lying  in  media  whose  absolute  indices  are  ftlt  /* „  p., .  .  .  ,  the  expres- 
sion pi  sl  +  p2  82  +  fi3  83  +  .  .  .  represents  the  time  of  passage.  This 
expression,  which  may  be  called  the  sum  of  such  terms  as  ps,  must 
therefore  fulfil  the  above  condition;  that  is  to  say,  the  points  of 
incidence  on  the  surfaces  of  separation  must  be  so  situated  that  this 
sum  either  remains  absolutely  constant  when  small  changes  are  sup- 
posed to  be  made  in  the  positions  of  these  points,  or  else  retains  that 
approximate  constancy  which  is  characteristic  of  maxima  and  minima. 
Conversely,  all  lines  from  a  luminous  point  which  fulfil  this  condition, 
will  be  paths  of  actual  rays. 

.  1085.  Terrestrial  Refraction.1 — The  atmosphere  may  be  regarded  as 
nomogeneous  when  we  confine  our  attention  to  small  portions  of  it, 
and  hence  it  is  sensibly  true,  in  ordinary  experiments  where  no  great 
distances  are  concerned,  that  rays  of  light  in  air  are  straight,  just  as 
it  is  true  in  the  same  limited  sense  that  the  surface  of  a  liquid  at  rest 
is  a  horizontal  plane.  The  surface  of  an  ocean  is  not  plane,  but 
approximately  spherical,  its  curvature  being  quite  sensible  in  ordinary 
nautical  observations,  where  the  distance  concerned  is  merely  that 
of  the  visible  sea-horizon;  and  a  correction  for  curvature  is  in  like 
manner  required  in  observing  levels  on  land.  If  the  observer  is 
standing  on  a  perfectly  level  plain,  and  observing  a  distant  object  at 
precisely  the  same  height  as  his  eye  above  the  plain,  it  will  appear 
to  be  below  his  eye,  for  a  horizontal  plane  through  his  eye  will  pass 
above  it,  since  a  perfectly  level  plain  is  not  plane,  but  snares  in  the 
general  curvature  of  the  earth.  It  is  easily  proved  that  the  apparent 
depression  due  to  this  cause  is  half  the  angle  between  the  verticals 
at  the  positions  of  the  observer  and  of  the  object  observed.  But 
experience  has  shown  that  this  apparent  depression  is  to  a  consider- 
able extent  modified  by  an  opposite  disturbing  cause,  called  terres- 
trial refraction.  When  the  atmosphere  is  in  its  normal  condition, 
a  ray  of  light  from  the  object  to  the  observer  is  not  straight,  but 
is  slightly  concave  downwards. 

This  curvature  of  a  nearly  horizontal  ray  is  not  due  to  the  curva- 
ture of  the  earth  and  of  the  layers  of  equal  density  in  the  earth's 
atmosphere,  as  is  often  erroneously  supposed,  but  would  still  exist, 

1  For  the  leading  idea  which  is  developed  in  §§  1085-1087,  the  Editor  is  indebted  to 
sxiggestions  from  Professor  James  Thomson. 


1106  WAVE   THEORY   OF  LIGHT. 

and  with  no  sensible  change  in  its  amount,  if  the  earth's  surface  were 
plane,  and  the  directions  of  gravity  everywhere  parallel.  It  is  due 
to  the  fact  that  light  travels  faster  in  the  rarer  air  above  than  in  the 
denser  air  below,  so  that  time  is  saved  by  deviating  slightly  to  the 
upper  side  of  a  straight  course.  The  actual  amount  of  curvature  (as 
determined  by  surveying)  is  from  |  to  •£$  of  the  curvature  of  the 
earth;  that  is  to  say,  the  radius  of  curvature  of  the  ray  is  from  2  to 
10  times  the  earth's  radius. 

1086.  Calculation  of  Curvature  of  Ray.  —  In  order  to  calculate  the 
radius  of  curvature  from  physical  data,  it  is  better  to  approach  the 
subject  from  a  somewhat  different  point  of  view. 

The  wave-  fronts  of  a  ray  in  air  are  perpendicular  to  the  ray;  and 
if  the  ray  is  nearly  horizontal,  its  wave-fronts  will  be  nearly  vertical. 
If  two  of  these  wave-fronts  are  produced  downwards  until  they  meet, 
the  distance  of  their  intersection  from  the  ray  will  be  the  radius  of 
curvature.  Let  us  consider  two  points  on  the  same  wave-front,  one 
of  them  a  foot  above  the  other;  then  the  upper  one  being  in  rarer  air 
will  be  advancing  faster  than  the  lower  one,  and  it  is  easily  shown 
that  the  difference  of  their  velocities  is  to  the  velocity  of  either,  as 
1  foot  is  to  the  radius  of  curvature. 

Put  p  for  the  radius  of  curvature  in  feet,  v  and  v+S  v  for  the  two 
velocities,  p.  and  p  —  $p  for  the  indices  of  refraction  of  the  air  at  the 
two  points.  Then  we  have 

I  =  SJL  =  SJ*  =  8  M  nearly.  (1) 

P         v         M 

Now  it  has  been  ascertained,  by  direct  experiment,  that  the  value 
of  ft,  —  I  for  air,  within  ordinary  limits  of  density,  is  sensibly  pro- 
portional to  the  density  (even  when  the  temperature  varies),  and  is 
•0002943  or  -srrw&t  the  density  corresponding  to  the  pressure  760mm, 
(at  Paris)  and  temperature  0°C.  The  difference  of  density  at  the  two 
points  considered,  supposing  them  both  to  be  at  the  same  tempera- 
ture, will  be  to  the  density  of  either  as  1  foot  is  to  the  "  height  of 
the  homogeneous  atmosphere"  in  feet,  which  call  H  (§  211).  Then 

-—  will  be  g,  and  the  value  of  -  in  (1)  may  be  written 


Hence  p  is  3400  times  the  height  of  the  homogeneous  atmosphere. 
But  this  height  is  about  5  miles,  or  -^  of  the  earth's  radius.     The 


ATMOSPHERIC   REFRACTION.  1107 

value  of  p  is  therefore  about  4J  radii  of  the  earth.  This  is  on  the 
assumptions  that  the  barometer  is  at  760mm,  the  thermometer  at  0°  C., 
and  that  there  is  no  change  of  temperature  in  ascending.  If  we  depart 
from  these  assumptions,  we  have  the  following  consequences: — 

I.  If  the  barometer  is  at  any  other  height,  the  factor  ^  remains 

unaltered,  and  the  other  factor  p  —  1  varies  directly  as  the  pressure. 

II.  If  the  temperature  is  t°  Centigrade,  H  is  changed  in  the  direct 
ratio  of  1  +  o  t,  a  denoting  the  coefficient  of  expansion.     The  first 

factor  g  is  therefore  changed  in  the  inverse  ratio  of  l+a£.     The 

second  factor  is  changed  in  the  same  ratio.  The  curvature  of  the  ray 
therefore  varies  inversely  as  (1  +  a  t)z. 

III.  Suppose  the  temperature  decreases  upwards  at  the  rate  of  - 

of  a  degree  Centigrade  per  foot.  The  expansion  due  to  ^-  of  a  degree 
Centigrade  is  ^.  The  first  factor  ^,  or  differe^[ydensity,  will 

therefore  become  H  -  2-3  n,  which,  if  we  put  71  =  540  (corresponding 
to  1°  Fahr.  in  300  feet),  and  reckon  H  as  26,000,  is  approximately 

266o6-i4^o6  or  fil1  ~  1}'  The  second  factor  of  the  expression  for  ± 
is  unaffected.  It  appears,  then,  that  decrease  of  temperature  upwards 
at  the  rate  of  1°  C.  in  540  feet,  or  1°  F.  in  300  feet  (which  is  the  gene- 
rally-received average),  makes  the  curvature  of  the  ray  five-sixths  of 
what  it  would  be  if  the  temperature  were  uniform.1 

Combining  this  correction  with  correction  II.,  it  appears  that,  with 
a  mean  temperature  of  10°  C.  or  50°  F.,  and  barometer  at  760mm,  the 
curvature  of  a  nearly  horizontal  ray  (taking  the  earth's  curvature  as 
unity)  is 


This  is  in  perfect  agreement  with  observation,  the  received  average 
(obtained  as  an  empirical  deduction  from  observation)  being  %  or  £. 
1087.  Curvature  of  Inclined  Rays. — Thus  far  we  have  been  treating 

of  nearly  horizontal  rays.    To  adapt  our  formula  for  -  ( (2)  §  1086)  to 
the  case  of  an  oblique  ray,  we  have  merely  to  multiply  it  by  cos  0, 

1  If  the  temperature  decreases  upwards  at  the  rate  of  1°  C.  in  n  feet,  or  1°  F.  in  n'  feet, 
the  first  factor  of  the  expression  for  -  (which  would  be  g  at  uniform  temperature)  becomes 


1108  WAVE   THEORY   OF  LIGHT. 

0  denoting  the  inclination  of  the  ray  to  the  horizontal,  or  the  inclina- 
tion of  the  wave-front  to  the  vertical.  For,  if  we  still  compare  two 
points  a  foot  apart,  on  the  same  wave-front,  and  in  the  same  vertical 
plane  with  each  other  and  with  the  ray,  their  difference  of  height 

will  be  the  product  of  1  foot  by  cos  0,  and  ^  will  therefore  be  less 
than  before  in  the  ratio  cos  0. 

Hence  it  can  be  shown  that  the  earth's  curvature,  so  far  from 
being  the  cause  of  terrestrial  refraction,  rather  tends  in  ordinary  cir- 
cumstances to  diminish  it,  by  increasing  the  average  obliquity  of  a 
ray  joining  two  points  at  the  same  level. 

The  general  formula  for  the  curvature  of  a  ray  (lying  in  a  vertical 
plane)  at  any  point  in  its  length,  may  be  written 


(3) 


n  denoting  the  number  of  feet  of  ascent  which  give  a  decrease  of  1°  C., 
and  n  the  number  of  feet  which  give  a  decrease  of  1°  F.  The  unit 
of  length  for  H  and  p  may  be  anything  we  please. 

1088.  Astronomical  Refraction. — Astronomical  refraction,  in  virtue 
of  which  stars  appear  nearer  the  zenith  than  they  really  are,  can  be 
reduced  to  these  principles;  but  it  is  simpler,  in  the  case  of  stars  not 
more  than  70°  or  80°  from  the  zenith,  to  regard  the  earth  and  the 
layers  of  equal  density  in  the  atmosphere  as  plane,  and  to  assume 
(§  993)  that  the  final  result  is  the  same  as  if  the  rays  from  the  star 
were  refracted  at  once  out  of  vacuum  into  the  horizontal  stratum  of 
air  in  which  the  observer's  eye  is  situated.    If  z  be  the  apparent  and 
z+h  the  true  zenith  distance,  we  shall  thus  have 

H  sin  2  =  sin  (z  +  h) 

—  sin  2  cos  h  +  cos  z  sin  A 
=  sin  z  +  A  cos  z,  nearly, 

whence 

h  =  (p  -  1)  tan  z. 

1089.  Mirage. — An  appearance,  as  of  water,  is  frequently  seen  in 
sandy  deserts,  where  the  soil  is  highly  heated  by  the  sun.     The 
observer  sees  in  the  distance  the  reflection  of  the  sky  and  of  terres- 
trial objects,  as  in  the  surface  of  a  calm  lake.     This  phenomenon, 
which  is  called  mirage,  is  explained  by  the  heating  and  consequent 
rarefaction  of  the  air  in  contact  with  the  hot  soil.     The  density, 


MIRAGE.  1109 

within  a  certain  distance  of  the  ground,  increases  upwards,  and  rays 
traversing  this  portion  of  the  air  are  bent  upwards  (Fig.  773),  in 
Accordance  with  the  general  rule  that  the  concavity  must  be  turned 
towards  the  denser  side.  Rays  which  were  descending  at  a  very 
slight  inclination  before  entering  this  stratum  of  air  may  have  their 
direction  so  much  changed  as  to  be  bent  up  to  an  observer's  eye,  and 
the  change  of  direction  will  be  greatest  for  those  rays  which  have 


Fig.  773,-Theory  of  Mirage. 


descended  lowest;  for  these  will  not  only  have  travelled  for  the 
greatest  distance  in  the  stratum,  but  will  also  have  travelled  through 
that  part  of  it  in  which  the  change  of  density  is  most  rapid.  Hence, 
if  we  trace  a  pencil  of  rays  from  the  observer's  eye,  we  shall  find 
that  those  of  them  which  lie  in  the  same  vertical  plane  crass  each 
other  in  traversing  this  stratum,  and  thus  produce  inverted  images. 
If  the  stratum  is  thin  in  comparison  with  the  height  of  the  observer's 
eye,  the  appearance  presented  will  be  nearly  equivalent  to  that  pro- 
duced by  a  mirror,  while  the  objects  thus  reflected  are  also  seen  erect 
by  higher  rays  which  have  not  descended  into  the  stratum  where  this 
action  occurs. 

A  kind  of  inverted  mirage  is  often  seen  across  masses  of  calm 
water,  and  is  called  looming;  images  of  distant  objects,  such  as  ships 
or  hills,  being  seen  in  an  inverted  position  immediately  over  the 
objects  themselves.  The  explanation  just  given  of  the  mirage  of  the 
desert  will  apply  to  this  phenomenon  also,  if  we  suppose  at  a  certain 
height,  greater  than  that  of  the  observer's  eye,  a  layer  of  rapid  tran- 
sition from  colder  and  denser  air  below  to  warmer  and  rarer  air 
above. 


1110  WAVE  THEORY   OF   LIGHT. 

An  appearance  similar  to  mirage  may  be  obtained  by  gently 
depositing  alcohol  or  methylated  spirit  upon  water  in  a  vessel  with 
plate-glass  sides.  The  spirit,  though  lighter,  has  a  higher  index  of 
refraction  than  the  water,  and  rays  traversing  the  layer  of  transition 
are  bent  upwards.  This  layer  accordingly  behaves  'like  a  mirror 
when  looked  at  very  obliquely  by  an  eye  above  it.1 

1090.  Curved  Bays  of  Sound.— The  reasoning  of  §§  1084, 1086  can 
be  applied,  with  a  slight  modification,  to  the  propagation  of  sound. 

Sound  travels  faster  in  warm  than  in  cold  air.  On  calm  sunny 
afternoons,  when  the  ground  has  become  highly  heated  by  the  sun's 
rays,  the  temperature  of  the  air  is  much  higher  near  the  ground  than 
at  moderate  heights;  hence  sound  bends  upwards,  and  may  thus 
become  inaudible  to  observers  at  a  distance  by  passing  over  their 
heads.  On  the  other  hand,  on  clear  calm  nights  the  ground  is  cooled 
by  radiation  to  the  sky,  and  the  layers  of  air  near  the  ground  are 
colder  than  those  above  them;  hence  sound  bends  downwards,  and 
may  thus,  by  arching  over  intervening  obstacles,  become  audible  at 
distant  points,  which  it  could  not  reach  by  rectilinear  propagation. 
This  influence  of  temperature,  which  was  first  pointed  out  by  Pro- 
fessor Osborne  Reynolds,  is  one  reason  why  sound  from  distant 
sources  is  better  heard  by  night  than  by  day. 

A  similar  effect  of  wind  had  been  previously  pointed  out  by  Pro- 
fessor Stokes.  It  is  well  known  that  sound  is  better  heard  with  the 
wind  than  against  it.  This  difference  is  due  to  the  circumstance 
that  wind  is  checked  by  friction  against  the  earth,  and  therefore 
increases  in  velocity  upwards.  Sound  travelling  with  the  wind, 
therefore,  travels  fastest  above,  and  sound  travelling  against  the 
wind  travels  fastest  below,  its  actual  velocity  being  in  the  former 
case  the  sum,  and  in  the  latter  the  difference,  of  its  velocity  in  still 
air  and  the  velocity  of  the  wind.  The  velocity  of  the  wind  is  so 
much  less  than  that  of  sound,  that  if  uniform  at  all  heights  its  influ- 
ence on  audibility  would  scarcely  be  appreciable. 

1091.  Calculation. — To  calculate  the  curvature  of  a  ray  of  sound 
due  to  variation  of  temperature  with  height,  we  may  employ,  as  in 
§  1086,  the  formula  £  =  -*  where  I  v  denotes, the  difference  of  velocity 
for  a  difference  of  1  foot  in  height.  The  value  of  v  varies  as  V  (1  +a  t), 
or  approximately  as  1  +  \  a  t,  t  denoting  temperature,  and  a  the  co- 

1  A  more,  complete  discussion  of  the  optics  of  mirage  will  be  found  in  two  papers  by 
the  editor  of  this  work  in  the  Philosophical  Mayazine  for  March  and  April,  1873,  and  in 
Nature  for  Nov.  19  and  26,  1874. 


REFRACTION   OF  SOUND.  1111 

efficient  of  expansion,  which  is  ^3.  Hence  if  the  velocity  at  0°  be 
denoted  by  1,  the  value  at  t°  will  be  denoted  by  1  +  |  at;  and  if  the 
temperature  varies  by  ^  of  a  degree  per  foot,  the  value  of  -£  at  tem- 
peratures near  zero  will  be  ^,  that  is,  g^.  and  the  radius  of  curva- 
ture will  be  5  46  n  feet.  This  calculation  shows  that  the  bending  is 
much  more  considerable  for  rays  of  sound  than  for  rays  of  light. 

1092.  Diffraction  Fringes. — When  a  beam  of  direct  sunlight  is 
admitted  into  a  dark  room  through  a  narrow  slit,  a  screen  placed  at 
any  distance  to  receive  it  will  show  a  line  of  white  light,  bordered 
with  coloured  fringes  which  become  wider  as  the  slit  is  narrowed. 
They  also  increase  in  width  as  the  screen  is  removed  further  off.  If 
they  are  viewed  through  a  piece  of  red  glass  which  allows  only  red 
rays  to  pass,  they  will  appear  as  a  succession  of  bands  alternately 
bright  and  dark. 

To  explain  their  origin,  we  shall  suppose  the  sun's  rays  (which  may 
be  reflected  from  an  external  mirror)  to  be  perpendicular  to  the  plane 
of  the  slit,1  so  that  the  wave-fronts  are  parallel  to  this  plane,  and  we 
shall,  in  the  first  instance,  confine  our  attention  to  light  of  a  particular 
wave-length;  for  example,  that  of  the  light  transmitted  by  the  red 
glass.  Then,  if  the  slit  be  uniform  through  its  whole  length,  the 
positions  of  the  bright  and  dark  bands  will  be  governed  by  the  fol- 
lowing laws: — 

1.  The  darkest  parts  will  be  at  points  whose  distances  from  the 
two  edges  of  the  slit  differ  by  an  exact  number  of  wave-lengths.  If 
the  difference  be  one  wave-length,  the  light  .which  arrives  at  any 
instant  from  different  parts  of  the  width  of  the  slit  is  in  all  possible 
phases,  and  the  resultant  of  the  whole  is  zero.  In  fact,  the  disturb- 
ance produced  by  the  nearer  half  of  the  slit  cancels  that  produced 
by  the  remoter  half.  If  the  difference  be  n  wave-lengths,  we  can 
divide  the  slit  into  n  parts,  such  that  the  effect  due  to  each  part  is 
thus  nil. 

•2»  The  brightest  parts  will  be  at  points  whose  distances  from  the 
two  edges  of  the  slit  differ  by  an  exact  number  of  wave-lengths  plus 

1  That  is,  to  the  plane  of  the  two  knife-edges  by  which  the  slit  is  bounded  This  condi- 
tion can  only  be  strictly  fulfilled  for  a  single  point  on  the  sun's  disc.  Every  point  on  the 
sun's  surface  sends  out  its  own  waves  as  an  independent  source ;  and  waves  from  one  point 
cannot  interfere  with  waves  from  another.  In  the  experiment  as  described  in  the  text 
the  fringes  due  to  different  parts  of  the  sun's  surface  are  all  produced  at  once  on  the 
screen,  and  overlap  each  other. 


1112  WAVE  THEORY  OF  LIGHT. 

a  half.  Let  the  difference  be  w  +  |;  then  we  can  divide  the  slit  into 
n  inefficient  parts  and  one  efficient  part,  this  latter  having  only  half 
the  width  of  one  of  the  others.1 

Each  colour  of  light  has  its  own  alternate  bands  of  brightness  and 
darkness,  the  distance  from  band  to  band  being  greatest  for  red  and 
least  for  violet.  The  superposition  of  all  the  bands  constitutes  the 
coloured  fringes  which  are  seen. 

This  experiment  furnishes  the  simplest  answer  to  the  objection 
formerly  raised  to  the  undulatory  theory,  that  light  is  not  able,  like 
sound,  to  pass  round  an  obstacle,  but  can  only  travel  in  straight  lines. 
In  this  experiment  light  does  pass  round  an  obstacle,  and  turns  more 
and  more  away  from  a  straight  line  as  the  slit  is  narrowed. 

When  the  slit  is  not  exceedingly  narrow,  the  light  sent  in  oblique 
directions  is  quite  insensible  in  comparison  with  the  direct  light,  and 
no  fringes  are  visible.  "We  have  reason  to  think  that  when  sound 
passes  through  a  very  large  aperture,  or  when  it  is  reflected  from  a 
large  surface  (which  amounts  nearly  to  the  same  thing),  it  is  hardly 
sensible  except  in  front  of  the  opening,  or  in  the  direction  of  reflection."2 

There  are  several  other  modes  of  producing  diffraction  fringes, 
which  our  limits  do  not  permit  us  to  notice.  We  proceed  to  describe 
the  mode  of  obtaining  a  pure  spectrum  by  diffraction. 

1093.  Diffraction  by  a  Grating. — If  a  piece  of  glass  is  ruled  with 
parallel  equidistant  scratches  (by  means  of  a  dividing  engine  and 
diamond  point)  at  the  rate  of  some  hundreds  or  thousands  to  the  inch, 
we  shall  find,  on  looking  through  it  at  a  slit  or  other  bright  line  (the 
glass  being  held  so  that  the  scratches  are  parallel  to  the  slit),  that  a 
number  of  spectra  are  presented  to  view,  ranged  at  nearly  equal 
distances,  on  both  sides  of  the  slit.  If  the  experiment  is  made  under 
favourable  circumstances,  the  spectra  will  be  so  pure  as  to  show  a 
number  of  Fraunhofer's  lines. 

Instead  of  viewing  the  spectra  with  the  naked  eye,  we  may  with 
advantage  employ  a  telescope,  focussed  on  the  plane  of  the  slit;  or 
we  may  project  the  spectra  on  a  screen,  by  first  placing  a  convex 


1  Each  element  of  the  length  of  the  slit  tends  to  produce  a  system  of  circular  rings  (the 
screen  being  supposed  parallel  to  the  plane  of  the  slit).    If  the  width  of  the  slit  is  uniform, 
these  systems  will  be  precisely  alike,  and  will  have  for  their  resultant  a  system  of  straight 
bands,  parallel  to  the  slit  and  touching  the  rings.     These  are  the  bands  described  in  the 
text.    Hence,  to  determine  the  illumination  of  any  point  of  the  screen,  it  is  only  necessary 
to  attend,  as  in  the  text,  to  the  nearest  points  of  the  two  edges  of  the  slit. 

2  Airy,  Undulalonj  Theory.     Art.  28. 


DIFFRACTION. 


1113 


lens  so  as  to  form  an  image  of  the  slit  (which  must  be  very  strongly 
illuminated)  on  the  screen,  and  then  interposing  the  ruled  glass  in 
the  path  of  the  beam. 

A  piece  of  glass  thus  ruled  is  called  a  grating.1  A  grating  for 
diffraction  experiments  consists  essentially  of  a  number  of  parallel 
strips  alternately  transparent  and  opaque. 

The  distance  between  the  "fixed  lines"  of  the  spectra,  and  the 
distance  from  one  spectrum  to  the  next,  are  found  to  depend  on  the 
distance  of  the  strips  measured  from  centre  to  centre,  in  other  words, 
on  the  number  of  scratches  to  the  inch,  but  not  at  all  on  the  relative 
breadths  of  the  transparent  and  opaque 
strips.  This  latter  circumstance  only 
affects  the  brightness  of  the  spectra. 

Diffraction  spectra  are  of  great  practi- 
cal importance — 

1.  As  furnishing  a  uniform  standard  of 
reference  in  the  comparison  of  spectra. 

2.  As   affording    the  most   accurate 
method  of  determining  the  wave-lengths 
of    the    different    elementary    rays   of 
light. 

1094.  Principle  of  Diffraction  Spectrum. 
—Let  GG  (Fig.  774)  be  a  grating,  re- 
ceiving light  from  an  infinitely2  distant 
point  lying  in  a  direction  perpendicular 
to  the  plane  of  the  grating,  so  that  the 
wave-fronts  of  the  incident  light  are 
parallel  to  this  plane.  Let  a  convex 
lens  L  be  placed  on  the  other  side  of  the 
grating,  and  let  its  axis  make  an  acute 
angle  0  with  the  rays  incident  on  the 
grating.  Then  the  light  collected  at 
its  principal  focus  F  consists  of  all  the 
light  incident  upon  the  lens  parallel  to 
its  axis.  Let  s  denote  the  distance 


774. 
Principle  of  Diffraction  Spectrum. 


1  Engraved  glass  gratings  of  sufficient  size  for  spectroscopic  purposes  (say  an  inch  square) 
are  extremely  expensive  and  difficult  to  procure.  Lord  Rayleigh  has  made  numerous 
photographic  copies  of  such  gratings,  and  the  copies  appear  to  be  equally  effective  with 
the  originals. 

8  It  is  not  necessary  that  the  source  should  be  infinitely  distant  (or  the  incident  rays 
parallel);  but  this  is  the  simplest  case,  and  the  most  usual  case  in  practice. 


1114  WAVE   THEORY   OF   LIGHT. 

between  the  rulings,  measured  from  centre  to  centre,  so  that  if,  for 
example,  there  are  1000  lines  to  the  inch,  s  will  be  j-gVo"  of  an  inch; 
and  suppose  first  that  s  sin  6  is  exactly  equal  to  the  wave-length  X  of 
one  of  the  elementary  kinds  of  light.  Then,  of  all  the  light  which 
falls  upon  the  lens  parallel  to  its  axis,  the  left-hand  portion  in  the 
figure  is  most  retarded  (having  travelled  farthest),  and  the  right-hand 
portion  least,  the  retardation,  in  comparing  each  transparent  interval 
with  the  next,  being  constant,  and  equal  to  s  sin  6,  as  is  evident  from 
an  inspection  of  the  figure.  Now,  for  the  particular  kind  of  light  for 
which  \  =  s  sin  6,  this  retardation  is  exactly  a  wave-length,  and  all 
the  transparent  intervals  send  light  of  the  same  phase  to  the  focus  F; 
so  that,  if  there  are  1000  such  intervals,  the  resultant  amplitude 
of  vibration  at  F  is  1000  times  the  amplitude  due  to  one  interval 
alone.  For  light  of  any  other  wave-length  this  coincidence  of  phase 
will  not  exist.  For  example,  if  the  difference  between  X  and  s  sin  0 
is  I01oo  ^>  the  difference  of  phase  between  the  lights  received  from 
the  1st  and  2d  intervals  will  be  T^OTT  ^>  between  the  1st  and  3d  10200  X, 
between  the  1st  and  501st  yVrnr  'V  or  jus^  na^  a  wave-length,  and 
so  on.  The  1st  and  501st  are  thus  in  complete  discordance,  as  are 
also  the  2d  and  502d,  &c.  Light  of  every  wave-length  except  one 
is  thus  almost  completely  destroyed  by  interference,  and  the  light 
collected  at  F  consists  almost  entirely  of  the  particular  kind  defined 
by  the  condition 

X  =  a  sin  0.  (1) 

The  purity  of  the  diffraction  spectrum  is  thus  explained. 

If  a  screen  be  held  at  F,  with  its  plane  perpendicular  to  the  prin- 
cipal axis,  any  point  on  this  screen  a  little  to  one  side  of  F  will 
receive  light  of  another  definite  wave-length,  corresponding  to  an- 
other direction  of  incidence  on  the  lens,  and  a  pure  spectrum  will 
thus  be  depicted  on  the  screen. 

1095.  Practical  Application.  —  In  the  arrangement  actually  em- 
ployed for  accurate  observation,  the  lens  LL  is  the  object-glass  of  a 
telescope  with  a  cross  of  spider-lines  at  its  principal  focus  F.  The 
telescope  is  first  pointed  directly  towards  the  source  of  light,  and  is 
then  turned  to  one  side  through  a  measured  angle  6.  Any  fixed  line 
of  the  spectrum  can  thus  be  brought  into  apparent  coincidence  with 
the  cross  of  spider-lines,  and  its  wave-length  can  be  computed  by 
the  formula  (1). 

The  spectrum  to  which  formula  (1)  relates  is  called  the  spectrum 
of  the  first  order. 


DIFFRACTION   SPECTRUM.  1115 

There  is  also  a  spectrum  of  the  second  order,  corresponding  to 
values  of  6  nearly  twice  as  great,  and  for  which  the  equation  is 

2  X  =  s  sin  6.  (2) 

For  the  spectrum  of  the  third  order,  the  equation  is 

S\  =  ssm6;  (3) 

and  so  on,  the  explanation  of  their  formation  being  almost  precisely 
the  same  as  that  above  given.  There  are  two  spectra  of  each  order, 
one  to  the  right,  and  the  other  at  the  same  distance  to  the  left  of 
the  direction  of  the  source.  In  Angstrom's  observations,1  which  are 
the  best  yet  taken,  all  the  spectra,  up  to  the  sixth  inclusive,  were 
observed,  and  numerous  independent  determinations  of  wave-length 
were  thus  obtained  for  several  hundred  of  the  dark  lines  of  the  solar 
spectrum. 

The  source  of  light  was  the  infinitely  distant  image  of  an  illumi- 
nated slit,  the  slit  being  placed  at  the  principal  focus  of  a  collimator, 
and  illuminated  by  a  beam  of  the  sun's  rays  reflected  from  a  mirror. 

The  purity  of  a  diffraction  spectrum  increases  with  the  number  of 
lines  on  the  grating  which  come  into  play,  provided  that  they  are 
exactly  equidistant;  and  may  therefore  be  increased  either  by  in- 
creasing the  size  of  the  grating,  or  by  ruling  its  lines  closer  together. 
The  gratings  employed  by  Angstrom  were  about  f  of  an  inch  square, 
the  closest  ruled  having  about  4500  lines,  and  the  widest  1500. 

As  regards  brightness,  diffraction  spectra  are  far  inferior  to  those 
obtained  by  prisms.  To  give  a  maximum  of  light,  the  opaque  inter- 
vals should  be  perfectly  opaque,  and  the  transparent  intervals  per- 
fectly transparent;  but  even  under  the  most  favourable  conditions, 
the  whole  light  of  any  one  of  the  spectra  cannot  exceed  about  TV 
of  the  light  which  would  be  received  by  directing  the  telescope  to 
the  slit.  The  greatest  attainable  intrinsic  brightness  in  any  part  of 
a  diffraction  spectrum  is  thus  not  more  than  -^  of  the  intrinsic 
brightness  in  the  same  part  of  a  prismatic  spectrum,  obtained  with 
the  same  slit,  collimator,  and  observing  telescope,  and  with  the  same 
angular  separation  of  fixed  lines.  The  brightness  of  the  spectra  partly 
depends  upon  the  ratio  of  the  breadths  of  the  transparent  and 
opaque  intervals.  In  the  case  of  the  spectra  of  the  first  order,  the 
best  ratio  is  that  of  equality,  and  equal  departures  from  equality  in 
opposite  directions  give  identical  results;  for  example, if  the  breadth 

1  Angstrom,  Recherches  sur  la  Spectre  solaire,     Upsal,  186S. 


1116  WAVE  THEORY   OF  LIGHT. 

of  the  transparent  intervals  is  to  the  breadth  of  the  opaque  either 
as  1 : 5  or  as  5  : 1,  it  can  be  shown  that  the  quantity  of  light  in  the 
first  spectrum  is  just  a  quarter  of  what  it  would  be  with  the  breadths 
equal. 

When  a  diffraction  spectrum  is  seen  with  the  naked  eye,  the 
cornea  and  crystalline  of  the  eye  take  the  place  of  the  lens  L  L,  and 
form  a  real  image  on  the  retina  at  F. 

1096.  Retardation  Gratings. — If,  instead  of  supposing  the  bars  of 
the  grating  to  be  opaque,  we  suppose  them  to  be  transparent,  but  to 
produce  a  definite  change  of  phase  either  by  acceleration  or  retarda- 
tion, the  spectra  produced  will  be  the  same  as  in  the  case  above 
discussed,  except  as  regards  brightness.     We  may  regard  the  effect 
as  consisting  of  the  superposition  of  two  exactly  coincident  sets  of 
spectra,  one  due  to  the  spaces  and  the  other  to  the  bars.     Any  one 
of  the  resultant  spectra  may  be  either  brighter  or  less  bright  than 
either  of  its  components,  according  to  the  difference  of  phase  between 
them.   If  the  bars  and  spaces  are  equally  transparent,  the  two  super- 
imposed spectra  will  be  equally  bright,  and  their  resultant  at  any 
part  may  have  any  brightness  intermediate  between  zero  and  four 
times  that  of  either  component. 

1097.  Reflection  Gratings. — Diffraction  spectra  can  also  be  obtained 
by  reflection  from  a  surface  of  speculum  metal  finely  ruled  with 
parallel  and  equidistant  scratches.     The  appearance  presented  is  the 
same  as  if  the  geometrical  image  of  the  slit  (with  respect  to  the 
grating  regarded  as  a  .plane  mirror)  were  viewed  through  the  grating 
regarded  as  transparent. 

1098.  Standard  Spectrum. — The  simplicity  of  the  law  connecting 
wave-length  with  position,  in  the  spectra  obtained  by  diffraction, 
offers,  a  remarkable  contrast  to  the  "irrationality"  of  the  dispersion 
produced  by  prisms.   Diffraction  spectra  may  thus  be  fairly  regarded 
as  natural  standards  of  comparison;  and,  in  particular,  the  limiting 
form  (if  we  may  so  call  it)  to  which  the  diffraction  spectra  tend, 
as  sin  0  becomes  small  enough  to  be  identified  with  0,  so  that  devia- 
tion becomes  simply  proportional  to  wave-length,  is  generally  and 
deservedly  accepted  by  spectroscopists  as  the  absolute  standard  of 
reference.     This  limiting  form  is  often  briefly  designated  as  "the 
diffraction  spectrum;"  it  differs   in   fact   to  a  scarcely  appreciable 
extent  from  the  first, 'or  even  the  second  and  third  spectra  furnished 
in  ordinary  cases  by  a  grating. 

The  diffraction  spectrum  differs  notably  from  prismatic  spectra  in 


WAVE-LENGTHS. 


1117 


the  much  greater  relative  extension  of  the  red  end.  Owing  to  this 
circumstance,  the  brightest  part  of  the  diffraction  spectrum  of  solar 
light  is  nearly  in  its  centre. 

The  first  three  columns  of  numbers  in  the  subjoined  table  indicate 
the  approximate  distances  between  the  fixed  lines  B,  D,  E,  F,  G  in 
certain  prismatic  spectra,  and  in  the  standard  diffraction  spectrum, 
the  distance  from  B  to  G  being  in  each  case  taken  as  1000: — 


B  to  D,     v    . 
DtoE,     .     . 
E  to  F,     .    . 
FtoG,     .    . 

Flint-glass. 
Angle  of  60'. 

Bisulphide  of 
Carbon. 
Angle  of  60*. 

Diffraction,  or 
Difference  of 
Wave-length. 

Difference  of 
Wave-frequency. 

220 
214 
192 
374 

194 
206 
190 
410 

381 
243 
160 
216 

278 
232 
184 
306 

1000                 1000 

1000 

1000 

In  the  standard  diffraction  spectrum,  deviation  is  simply  propor- 
tional to  wave-length,  and  therefore  the  distance  between  two  colours 
represents  the  difference  of  their  wave-lengths.  It  has  been  sug- 
gested that  a  more  convenient  reference-spectrum  would  be  con- 
structed by  assigning  to  each  colour  a  deviation  proportional  to  its 
wave-frequency  (or  to  the  reciprocal  of  its  wave-length),  so  that  the 
distance  between  two  colours  will  represent  the  difference  between 
their  wave-frequencies.  The  result  of  thus  disposing  the  fixed  lines 
is  shown  in  the  last  column  of  the  above  table.  It  differs  from  pris- 
matic spectra  in  the  same  direction,  but  to  a  much  less  extent  than 
the  diffraction  spectrum. 

It  has  been  suggested  by  Mr.  Stoney  as  extremely  probable,  that 
the  bright  lines  of  spectra  are  in  many  cases  harmonics  of  some  one 
fundamental  vibration.  Three  of 'the  four  bright  lines  of  hydrogen 
have  wave-frequencies  exactly  proportional  to  the  numbers  20, 27,  and 
32;  and  in  the  spectrum  of  chloro-chromic  acid  all  the  lines  whose 
positions  have  been  observed  (31  in  number)  have  wave-frequencies 
which  are  multiples  of  one  common  fundamental. 

1099.  Wave-lengths. — Wave-lengths  of  light  are  commonly  stated 
in  terms  of  a  unit  of  which  1010  make  a  metre, — hence  called  the 
tenth-metre.  The  following  are  the  wave-lengths  of  some  of  the 
principal  "fixed  lines"  as  determined  by  Angstrom:1 — 

1  The  wave-lengths  of  the  spectral  lines  of  all  elementary  substances  will  be  found  in 
Dr.  \V.  M.  Watts'  Index  of  Spectra;  and  the  wave-lengths  and  wave-frequencies  of  the 
dark  lines  in  the  solar  spectrum,  with  the  names  of  the  substances  to  which  many  of  them 
are  due,  will  be  found  in  the  British  Association  fieport  for  1878  (Dublin),  pp.  40-91. 


1118 


WAVE  THEORY  OF  LIGHT. 


WAVE-LENGTHS   IN   TENTH-METRES. 


7604 

6867 
6562 
5895 
5889 


E 
F 
G 

Ht 

H, 


£269 
4861 
4307 
3963 
3933 


The  velocity  of  light  is  300  million  metres  per  second,  or  300  X  1016 
tenth-metres  per  second.  The  number  of  waves  per  second  for  any 
colour  is  therefore  300  x  1 016  divided  by  its  wave-length  as  above 
expressed.  Hence  we  find  approximately: — 


For  A. 
„  D. 
»  H. 


395  millions  of  millions  per  second. 
510         „  „  „ 

7CO 


1100.  Colours  of  Thin  Films.  Newton's  Rings.— If  two  pieces  of 
glass,  with  their  surfaces  clean,  are  brought  into  close  contact,  coloured 
fringes  are  seen  surrounding  the  point  where  the  contact  is  closest. 
They  are  best  seen  when  light  is  obliquely  reflected  to  the  eye  from  the 
surfaces  of  the  glass,  and  fringes  of  the  complementary  colours  may 
be  seen  by  transmitted  light.  A  drop  of  oil  placed  on  the  surface  of 
clean  water  spreads  out  into  a  thin  film,  which  exhibits  similar 
fringes  of  colour;  and  in  general,  a  very  thin  film  of  any  transparent 
substance,  separating  media  whose  indices  of  refraction  are  different 
from  its  own,  exhibits  colour,  especially  when  viewed  by  obliquely 
reflected  light.  In  the  first  experiment  above-mentioned,  the  thin 
film  is  an  air-film  separating  the  pieces  of  glass.  In  soap-bubbles  or 
films  of  soapy  water  stretched  on  rings,  a  similar  effect  is  produced 
by  a  small  thickness  of  water  separating  two  portions  of  air. 

The  colours,  in  all  these  cases,  when  seen  by  reflected  light,  are 
produced  by  the  mutual  interference  of  the  light  reflected  from  the 
two  surfaces  of  the  thin  film.  An  incident  ray  undergoes,  as  ex- 
plained in  §  992,  a  series  of  reflections  and  refractions;  and  we  may 
thus  distinguish,  for  light  of  any  given  refrangibility,  several  systems 
of  waves,  all  of  which  originally  came  from  the  same  source.  These 
systems  give  by  their  interference  a  series  of  alternately  bright  and 
dark  fringes;  and  when  ordinary  white  light  is  employed,  the  fringes 
are  broadest  for  the  colours  of  greatest  wave-length.  Their  super- 
position thus  produces  the  observed  colours.  The  colours  seen  by 
transmitted  light  may  be  similarly  explained. 

The  first  careful  observations  of  these  coloured  fringes  were  made 
by  Newton,  and  they  are  generally  known  as  Newton's  rings. 


CONCAVE   GRATINGS. 


1119 


1100  A.  Concave  Gratings.  —  Great  progress  has  been  made  in 
recent  years  in  the  manufacture  of  reflection  gratings  ruled  on 
speculum  metal.  Mr.  Rutherfurd  of  New  York  has  produced 
several  specimens  (containing  thirty  thousand  lines  in  a  space  of 
about  an  inch  and  three  quarters),  which  give  better  results  than 
have  ever  been  obtained  by  means  of  prisms.  Professor  Rowland 
of  Baltimore  has  improved  upon  these,  and  constructed  gratings 
with  160,000  lines  in  a  space  of  about  six  inches. 

Professor  Rowland  has  also  introduced  the  novelty  of  ruling 
gratings  on  concave  spherical  surfaces,  all  previous  gratings  having 
been  ruled  on  plane  surfaces.  He  is  thus  enabled  to  dispense  both 
with  a  collimating  lens  and  with  the  object-glass  of  the  observing 
telescope.  As  this  invention  promises  to  be  very  important,  we  shall 
explain  it  at  some  length. 

The  rule  which  determines  the  kind  of  light  that  will  be  thrown 

in  a  given  direction  by  any  part 
of  a  reflection  grating  is,  that 
the  total  length  of  path  from 
the  slit  to  a  point  lying  in  this 
direction  differs  by  one  wave- 
length or  an  exact  number  of 
wave-lengths  of  this  particular 
light,  as  we  pass  from  one  bar 
to  the  next.  If  the  slit  is  at  the 
same  distance  from  all  the  bars, 
the  difference  of  path  will  be  the 
difference  of  the  two  reflected 
rays,  and  will  be  the  projection 
of  the  distance  between  the  bars 
on  one  of  these  rays;  so  that,  as  in  §1094,  we  shall  have  8  sin  8  equal 
to  X  or  a  multiple  of  A,  0  now  denoting  the  angle  of  reflection. 

In  Fig.  774A,  O  is  the  centre  of  the  sphere  of  which  the  grating 
forms  part,  A  the  middle  point  of  the  grating,  B  A  C  a  section  of 
the  grating  perpendicular  to  the  rulings. 

First  let  the  slit  be  at  O.  For  the  spectra  of  the  first  order  the 
angle  of  reflection  0  for  a  given  kind  of  light  is  determined  by  the 

equation 

s  sin  9  =  X, 

and  if,  round  O  as  centre,  we  describe  a  circle  of  radius  O  A  sin  0, 
this  circle  will  be  touched  by  all  the  reflected  rays.     An  arc  b  a  c 
71 


rig.  774A.-concaye 


1120*  WAVE  THEOEY   OF  LIGHT. 

of  this  circle  containing  the  same  number  of  degrees  as  the  grating 
BAG  will  be  the  caustic.  The  narrowest  part  of  the  sheaf  of  rays 
will  be  at  a  the  middle  point  of  this  arc,  being  the  point  where  it  is 
touched  by  the  ray  A  a  from  the  middle  point  of  the  grating.  Since 
O  a  if  joined  is  at  right  angles  to  A  a,  the  point  a  lies  on  the  circle 
described  on  O  A  as  diameter — the  dotted  circle  in  the  figure.  For 
the  spectra  of  the  second  order,  0  is  determined  by  the  equation 

s  sin  0  =  2  X, 

and  the  caustic  will  be  the  circular  arc  b'a'c'  described  about  0  as 
centre  with  a  radius  double  of  the  radius  of  b  a  c,  the  construction 
being  in  other  respects  the  same;  and  similar  reasoning  applies  to 
the  spectra  of  higher  orders. 

All  the  caustics  of  eVery  order  and  for  every  value  of  X  will  thus 
have  their  brightest  points  on  the  circle  described  on  O  A  as  diameter. 

If  the  slit  be  at  a,  the  reflected  rays  of  the  first  order  will  form 
one  caustic  at  0  and  another  at  a;  for  the  rays  incident  from  a  on 
consecutive  bars  of  the  grating  at  A  will  have  a  common  difference 
of  X,  and  the  distances  of  these  bars  from  a'  have  a  common  differ- 
ence of  2  X. 

Neglecting  the  breadth  of  the  reflected  beam  at  its  narrowest 
part,  we  may  regard  a  and  a'  as  foci  conjugate  to  O,  and  may  regard 
the  dotted  circle  as  the  locus  of  a  point  from  which  rays  to  all  parts 
of  the  arc  BAG  have  the  same  angle  of  incidence;  whence  it  follows 
that,  if  the  slit  be  anywhere  on  this  circle,  all  the  spectra  will  focus 
themselves  along  the  circle.  Hence  the  grating,  the  slit,  and  the 
screen  for  receiving  the  spectra,  or  the  eye-piece  for  viewing  them, 
may  be  fixed  at  the  ends  of  three  equal  arms  all  pivoted  at  the 
centre  of  this  circle. 

A  pencil  of  rays  from  a  single  point  at  O  will  not  meet  in  a  single 
point  at  a,  even  if  we  regard  the  breadth  of  the  beam  at  a  as 
negligible,  but  in  a  focal  line  perpendicular  to  the  plane  of  the 
diagram,  and  they  will  meet  again  in  a  second  focal  line  ef  in  the 
plane  of  the  diagram.  The  lines  of  the  spectrum  due  to  a  slit  at  O, 
perpendicular  to  the  plane  of  the  diagram,  will  therefore  be  focussed 
at  a,  while  the  transverse  lines  due  to  particles  of  dust  in  the  slit 
will  be  focussed  at  ef.  Similar  remarks  apply  to  the  spectra  of  the 
second  and  higher  orders.  Hence  the  spectra  as  actually  observed 
with  a  concave  grating  have  the  advantage  of  not  showing  dust  lines. 


CHAPTER    LXXV. 


POLARIZATION  AND  DOUBLE   REFRACTION. 


1101.  Polarization. — When  a  piece  of  the  semi-transparent  mineral 
called  tourmaline  is  cut  into  slices  by  sections  parallel  to  its  axis,  it  is 
found  that  two  of  these  slices,  if  laid  one  upon  the  other  in  a  particular 
relative  position,  as  A,  B  (Fig.  775),  form  an  opaque  combination. 
Let  one  of  them,  in  fact,  be  turned  round  upon  the  other  through 
various  angles  (Fig.  775).  It  will  be  found  that  the  combination  is 
most  transparent  in  two  posi- 
tions differing  by  180°,  one  of 
them  a  b  being  the  natural 
position  which  they  originally 
occupied  in  the  crystal;  and 

that  it  is  most  Opaque  in  the  jig.  775. -Tourmaline  Plates. 

two  positions  at  right  angles 

to  these.  It  is  not  necessary  that  the  slices  should  be  cut  from  the 
same  crystal.  Any  two  plates  of  tourmaline  with  their  faces  parallel 
to  the  axis  of  the  crystals  from  which  they  were  cut,  will  exhibit 
the  same  phenomenon.  The  experiment  shows  that  light  which  has 
passed  through  one  such  plate  is  in  a  peculiar  and  so  to  speak  unsym- 
metrical  condition.  It  is  said  to  be  plane-polarized.  According  to 
the  undulatory  theory,  a  ray  of  common  light  contains  vibrations  in 
all  planes  passing  through  the  ray,  and  a  ray  of  plane-polarized 
light  contains  vibrations  in  one  plane  only.  Polarized  light  cannot 
be  distinguished  from  common  light  by  the  naked  eye;  and  for  all 
experiments  in  polarization  two  pieces  of  apparatus  must  be  employed 
— one  to  produce  polarization,  and  the  other  to  show  it.  The  former 
is  called  the  polarizer,  the  latter  the  analyser;  and  every  apparatus 
that  serves  for  one  of  these  purposes  will  also  serve  for  the  other. 
In  the  experiment  above  described,  the  plate  next  the  eye  is  the 


1120 


POLARIZATION   AND   DOUBLE  REFRACTION. 


analyser.  The  usual  process  in  examining  light  with  a  view  to  test 
whether  it  is  polarized,  consists  in  looking  at  it  through  an  analyser, 
and  observing  whether  any  change  of  brightness  occurs  as  the  analyser 
is  rotated.  When  the  light  of  the  blue  sky  is  thus  examined,  a 
difference  of  brightness  can  always  be  detected  according  to  the 
position  of  the  analyser,  especially  at  the  distance  of  about  90°  from 
the  sun.  In  all  such  cases  there  are  two  positions  differing  by  180°, 
which  give  a  minimum  of  light,  and  the  two  positions  intermediate 
between  these  give  a  maximum  of  light. 

The  extent  of  the  changes  thus  observed  IB  a  measure  of  the  com- 
pleteness of  the  polarization  of  the  light. 

1102.  Polarization  by  Reflection. — Transmission  through  tourmaline 
is  only  one  of  several  ways  in  which  light  can  be  polarized.  When 
a  beam  of  light  is  reflected  from  a  polished  surface  of  glass,  wood, 
ivory,  leather,  or  any  other  non-metallic  substance,  at  an  angle  of 
from  50°  to  60°  with  the  normal,  it  is  more  or  less  polarized,  and  in 
like  manner  a  reflector  composed  of  any  of  these  substances  may  be 
employed  as  an  analyser.  In  so  using  it,  it  should  be  rotated  about 
an  axis  parallel  to  the  incident  rays  which  are  to  be  tested,  and  the 
observation  consists  in  noting  whether  this  rotation  produces  changes 
in  the  amount  of  reflected  light. 

Mains'  Polariscope  (Fig.  776)  consists  of  two  reflectors  A,  B,  one 

serving  as  polarizer  and  the  other 
as  analyser,  each  consisting  of  a 
pile  of  glass  plates.  Each  of  these 
reflectors  can  be  turned  about  a 
horizontal  axis;  and  the  upper  one 
(which  is  the  analyser)  can  also 
be  turned  about  a  vertical  axis, 
the  amount  of  rotation  being  mea- 
sured on  the  horizontal  circle  C  C. 
To  obtain  the  most  powerful  ef- 
fects, each  of  the  reflectors  should 
be  set  at  an  angle  of  about  33°  to 
the  vertical,  and  a  strong  beam  of 
common  light  should  be  allowed 

to  fall  upon  the  lower  pile  in  such  a  direction  as  to  be  reflected 
vertically  upwards.  It  will  thus  fall  upon  the  centre  of  the  upper 
pile,  and  the  angles  of  incidence  and  reflection  on  both  the  piles  will 
be  about  57°.  The  observer  looking  into  the  upper  pile,  in  such  a 


Fig.  776.— Malus'  Polariscope. 


POLARIZATION   BY   REFLECTION.  1121 

direction  as  to  receive  the  reflected  beam,  will  find  that,  as  the  upper 
pile  is  rotated  about  a  vertical  axis,  there  are  two  positions  (differing 
by  180°)  in  which  he  sees  a  black  spot  in  the  centre  of  the  field  of 
view,  these  being  the  positions  in  which  the  upper  pile  refuses  to 
reflect  the  light  reflected  to  it  from  the  lower  pile.  They  are  90°  on 
either  side  of  the  position  in  which  the  two  piles  are  parallel;  this 
latter,  and  the  position  differing  from  it  by  180°,  being  those  which 
give  a  maximum  of  reflected  light. 

For  every  reflecting  substance  there  is  a  particular  angle  of  inci- 
dence which  gives  a  maximum  of  polarization  in  the  reflected  light. 
It  is  called  the  polarizing  angle  for  the  substance,  and  its  tangent  is 
always  equal  to  the  index  of  refraction  of  the  substance;  or  what 
amounts  to  the  same  thing,  it  is  that  particular  angle  of  incidence 
which  is  the  complement  of  the  angle  of  refraction,  so  that  the 
refracted  and  reflected  rays  are  at  right  angles.1  This  important  law 
was  discovered  experimentally  by  Sir  David  Brewster. 

The  reflected  ray  under  these  circumstances  is  in  a  state  of  almost 
complete  polarization;  and  the  advantage  of  employing  a  pile  of 
plates  consists  merely  in  the  greater  intensity  of  the  reflected  light 
thus  furnished.  The  transmitted  light  is  also  polarized;  it  diminishes 
in  intensity,  but  becomes  more  completely  polarized,  as  the  number 
of  plates  is  increased.  The  reflected  and  the  transmitted  light  are 
in  fact  mutually  complementary,  being  the  two  parts  into  which 
common  light  has  been  decomposed;  and  their  polarizations  are 
accordingly  opposite,  so  that,  if  both  the  transmitted  and  reflected 
beams  are  examined  by  a  tourmaline,  the  maxima  of  obscuration 
will  be  obtained  by  placing  the  axis  of  the  tourmaline  in  the 
one  case  parallel  and  in  the  other  perpendicular  to  the  plane  of 
incidence. 

It  is  to  be  noted  that  what  is  lost  in  reflection  is  gained  in  trans- 
mission, and  that  polarization  never  favours  reflection  at  the  expense 
of  transmission. 

1103.  Plane  of  Polarization. — That  particular  plane  in  which  a  ray 
of  polarized  light,  incident  at  the  polarizing  angle,  is  most  copiously 
reflected,  is  called  the  plane  of  polarization  of  the  ray.  When  the 
polarization  is  produced  by  reflection,  the  plane  of  reflection  is  the 

1  Adopting  the  indices  of  refraction  given  in  the  table  §  986,  we  find  the  following  values 
for  the  polarizing  angle  for  the  undermentioned  substances : — 


Diamond,  ...     67°  43'  to  70°    3' 
Flint-glass,     .    .     57°  36' to  58°  40' 


Crown-glass,  .    .     56°  51'  to  57°  23' 
Pure  Water, 53°  11' 


1122        POLARIZATION  AND  DOUBLE  REFRACTION. 

plane  of  polarizati6n.  According  to  Fresnel's  theory,  which  is  that 
generally  received,  the  vibrations  of  light  polarized  in  any  plane  are 
perpendicular  to  that  plane  (§  1115).  The  vibrations  of  a  ray  reflected 
at  the  polarizing  angle  are  accordingly  to  be  regarded  as  perpendi- 
cular to  the  plane  of  incidence  and  reflection,  and  therefore  as  parallel 
to  the  reflecting  surface. 

1104.  Polarization  by  Double  Refraction. — We  have  described  in 
§  998  some  of  the  principal  phenomena  of  double  refraction  in  uniaxal 
crystals.     We  have  now  to  mention  the  important  fact  that  the  two 
rays  furnished  by  double  refraction  are  polarized,  the  polarization  in 
this  case  being  more  complete  than  in  any  of  the  cases  thus  far  dis- 
cussed.   On  looking  at  the  two  images  through  a  plate  of  tourmaline, 
or  any  other  analyser,  it  will  be  found  that  they  undergo  great  varia- 
tions of  brightness  as  the  analyser  is  rotated,  one  of  them  becoming 
fainter  whenever  the  other  becomes  brighter,  and  the  maximum 
brightness  of  either  being  simultaneous  with  the  absolute  extinction 
of  the  other.  If  a  second  piece  of  Iceland-spar  be  used  as  the  analyser, 
four  images  will  be  seen,  of  which  one  pair  become  dimmer  as  the 
other  pair  become  brighter,  and  either  of  these  pairs  can  be  extin- 
guished by  giving  the  analyser  a  proper  position. 

1105.  Theory  of  Double  Refraction. — The  existence  of  double  refrac- 
tion admits  of  a  very  natural  explanation  on  the  undulatory  theory. 
In  uniaxal  crystals  it  is  assumed  that  the  elasticity  of  the  luminifer- 
ous  aether  is  the  same  for  all  vibrations  executed  in  directions  perpen- 
dicular to  the  axis;  and  that,  for  vibrations  in  other  directions,  the 
elasticity  varies  solely  according  to  the  inclination  of  the  direction 
of  vibration  to  the  axis.     There  are  two  classes  of  doubly-refracting 
uniaxal  crystals,  called  respectively  positive  and  negative.     In  the 
former  the  elasticity  for  vibrations  perpendicular  to  the  axis  is  a 
maximum;  in  the  latter  it  is  a  minimum.     Iceland-spar  belongs  to 
the  latter  class;  and  as  small  elasticity  implies  slow  propagation,  a 
ray  propagated  by  vibrations  perpendicular  to  the  axis  will,  in  this 
crystal,  travel  with  minimum  velocity;  while  the  most  rapid  pro- 
pagation will  be  attained  by  rays  whose  vibrations  are  parallel  to 
the  axis. 

Consider  any  plane  oblique  to  the  axis.  Through  any  point  in 
this  plane  we  can  draw  one  line  perpendicular  to  the  axis;  and  the 
line  at  right  angles  to  this  will  have  smaller  inclination  to  the  axis 
than  any  other  line  in  the  plane.  These  two  lines  are  the  directions 
of  least  and  greatest  resistance  to  vibration;  the  former  is  the  direc- 


THEORY  OF  DOUBLE  REFRACTION.  1123 

tion  of  vibration  for  an  ordinary,  and  the  latter  for  an  extraordinary 
ray.  The  velocity  of  propagation  is  the  same  for  the  ordinary  rays 
in  all  directions  in  the  crystal,  so  that  the  wave-surface  for  these  is 
spherical ;  but  the  velocity  of  propagation  for  the  extraordinary  rays 
differs  according  to  their  inclination  to  the  axis,  and  their  wave- 
surface  is  a  spheroid  whose  polar  diameter  is  equal  to  the  diameter 
of  the  aforesaid  sphere.  The  sphere  and  spheroid  touch  one  another 
at  the  extremities  of  this  diameter  (which  is  parallel  to  the  axis  of 
the  crystal),  and  the  ordinary  and  extraordinary  rays  coincide  both 
in  direction  and  velocity  along  this  common  diameter.  The  general 
construction  for  the  path  of  the  extraordinary  ray  is  due  to  Huygens, 
and  has  been  described  in  §  1081,  Fig.  770,  where  C  A  is  the  incident 
and  A  F  the  refracted  ray. 

When  the  plane  of  incidence  contains  the  axis,  the  spheroid  will 
be  symmetrical  with  respect  to  this  plane;  and,  therefore,  when  we 
draw  the  tangent  plane  E  F  perpendicular  to  the  plane  of  incidence 
(as  directed  in  the  construction)  the  point  of  contact  F  will  lie  in  the 
plane  of  incidence. 

Another  special  case  is  that  in  which  the  plane  of  incidence  is 
perpendicular  to  the  axis,  and  the  refracting  surface  parallel  to  the 
axis.  In  this  case  also  the  spheroid  will  be  symmetrical  with  respect 
to  the  plane  of  incidence,  which  will  in  fact  be  the  equatorial  plane 
of  the  spheroid,  and  the  point  of  contact  F  will  as  before  lie  in  this 
plane.  Moreover  since  the  section  is  equatorial  it  is  a  circle,  and 
hence,  as  shown  in  Fig.  771,  the  law  of  sines  will  be  applicable.  Th^ 
ratio  of  the  sines  of  the  angles  of  incidence  and  refraction  for  this 
particular  case  is  called  the  extraordinary  index  of  refraction  for 
the  crystal.  It  is  the  ratio  of  the  velocity  in  air  to  the  velocity 
along  an  equatorial  radius  of  the  spheroid. 

In  general,  the  spheroid  is  not  symmetrical  with  respect  to  the 
plane  of  incidence,  and  the  refracted  ray  A  F  does  not  lie  in  this 
plane. 

Tourmaline,  like  Iceland-spar,  is  a  negative  uniaxal  crystal;  and 
its  use  as  a  polarizer  depends  on  the  property  which  it  possesses  of 
absorbing  the  ordinary  much  more  rapidly  than  the  extraordinary 
ray,  so  that  a  thickness  which  is  tolerably  transparent  to  the  latter 
is  almost  completely  opaque  to  the  former. 

1106.  Nicol's  Prism. — One  of  the  most  convenient  and  effective 
contrivances  for  polarizing  light,  or  analysing  it  when  polarized,  is 
that  known,  from  the  name  of  its  inventor,  as  Nicol's  prism.  It  is 


1124 


POLARIZATION  AND   DOUBLE  REFRACTION. 


Fig.  777.— Nicol's  Prism. 


made  by  slitting  a  rhomb  of  Iceland-spar  along  a  diagonal  plane 

acbd  (Fig.  777),  and  cementing  the  two  pieces  together  in  their 

natural    position    by   Canada    balsam,  a 

•*\  substance  whose  refractive  index  is  inter- 

mediate between  the  ordinary  and  extra- 
ordinary indices  of  the  crystal.1  A  ray  of 
common  light  S I  undergoes  double  refrac- 
tion on  entering  the  prism.  Of  the  two 
rays  thus  formed,  the  ordinary  ray  is 
totally  reflected  on  meeting  the  first  sur- 
face of  the  balsam,  and  passes  out  at  one 
side  of  the  crystal,  as  o  0 ;  while  the  ex- 
traordinary ray  is  transmitted  through 
the  balsam  as  through  a  parallel  plate, 
and  finally  emerges  at  the  end  of  the 
prism,  in  the  direction  e  E,  parallel  to  the 
original  direction  S  I.  This  apparatus  has 
nearly  all  the  convenience  of  a  tourmaline 

plate,  with  the  advantages  of  much  greater  transparency  and  of 
complete  polarization. 

In  Foucault's  prism,  which  is  extensively  used  instead  of  Nicol's, 
the  Canada  balsam  is  omitted,  and  there  is  nothing  but  air  between 
the.  two  pieces.  This  change  has  the  advantage  of  shortening  the 
prism  (because  the  critical  angle  of  total  reflection  depends  on  the 
relative  index  of  refraction  of  the  two  media),  but  gives  a  smaller 
field  of  view,  and  rather  more  loss  of  light  by  reflection. 

1107.  Colours  produced  by  Elliptic  Polarization. — Very  beautiful 
colours  may  be  produced  by  the  peculiar  action  of  polarized  light. 
For  example,  if  a  piece  of  selenite  (crystallized  gypsum)  about  the 
thickness  of  paper,  is  introduced  between  the  polarizer  and  analyser 
of  any  polarizing  arrangement,  and  turned  about  into  different 
directions,  it  will  in  some  positions  appear  brightly  coloured,  the 
colour  being  most  decided  when  the  analyser  is  in  either  of  the  two 
critical  positions  which  give  respectively  the  greatest  light  and  the 
greatest  darkness.  The  colour  is  changed  to  its  complementary  by 

1  a  and  b  are  the  corners  at  which  three  equal  obtuse  angles  meet  (§  999).  The  ends  of 
the  rhomb  which  are  shaded  in  the  figure  are  rhombuses.  Their  diagonals  drawn  through 
a  and  b  respectively  will  lie  in  one  plane,  which  will  contain  the  axis  of  the  crystal,  and 
will  cut  the  plane  of  section  a  c  b  d  at  right  angles.  The  length  of  the  rhomb  is  about  three 
and  a  half  times  its  breadth. 


COLOUR  BY   ELLIPTIC   POLARIZATION.  1125 

rotating  the  analyser  through  a  right  angle;  but  rotation  of  the  piece 
of  selenite,  when  the  analyser  is  in  either  of  the  critical  positions, 
merely  alters  the  depth  of  the  colour  without  changing  its  tint,  and 
in  certain  critical  positions  of  the  selenite  there  is  a  complete  absence 
of  colour.  Thicker  plates  of  selenite  restore  the  light  when  ex- 
tinguished by  the  analyser,  but  do  not  show  colour. 

1108.  Explanation. — The  following  is  the  explanation  of  these 
appearances.  Let  the  analyser  be  turned  into  such  a  position  as  to 
produce  complete  extinction  of  the  plane-polarized  light  which  comes 
to  it  from  the  polarizer;  and  let  the  plane  of  polarization  and  the 
plane  perpendicular  thereto  (and  parallel  to  the  polarized  rays)  be 
called  the  two  planes  of  reference.  Let  the  slice  of  selenite  be  laid 
so  that  the  polarized  rays  pass  through  it  normally.  Then  there  are 
two  directions,  at  right  angles  to  each  other,  which  are  the  directions 
of  greatest  and  least  elasticity  in  the  plane  of  the  slice.  Unless  the 
slice  is  laid  so  that  these  directions  coincide  with  the  two  planes  of 
reference,  the  plane-polarized  light  which  is  incident  upon  it  will  be 
broken  up  into  two  rays,  one  of  which  will  traverse  it  more  rapidly 
than  the  other.  Referring  to  the  diagram  of  Lissajous'  figures  (Fig. 
634),  let  the  sides  of  the  rectangle  be  the  directions  of  greatest  and 
least  elasticity,  and  let  the  diagonal  line  in  the  first  figure  be  the 
direction  of  the  vibrations  of  an  incident  ray, — this  diagonal  accord- 
ingly lies  in  one  of  the  two  planes  of  reference.  In  traversing  the 
slice,  the  component  vibrations  in  the  directions  of  greatest  and 
least  elasticity  will  be  propagated  with  unequal  velocities;  and  if  the 
incident  ray  be  homogeneous,  the  emergent  light  will  be  elliptically 
polarized;  that  is  to  say,  its  vibrations,  instead  of  being  rectilinear, 
will  be  elliptic,  precisely  on  the  principle1  of  Blackburn's  pendulum 
(§  924).  The  shape  of  the  ellipse  depends,  as  in  the  case  of  Lis- 
sajous' figures,  on  the  amount  of  retardation  of  one  of  the  two  com- 
ponent vibrations  as  compared  with  the  other,  and  this  is  directly 
proportional  to  the  thickness  of  the  slice.  The  analyser  resolves 
these  elliptic  vibrations  into  two  rectilinear  components  parallel 
and  perpendicular  to  the  original  direction  of  vibration,  and 
suppresses  one  of  these  components,  so  that  only  the  other  remains. 

1  The  principle  is  that,  whereas  displacement  of  a  particle  parallel  to  either  of  the  sides 
of  the  rectangle  calls  out  a  restoring  force  directly  opposite  to  the  displacement,  displace- 
ment in  any  other  direction  calls  out  a  restoring  force  inclined  to  the  direction  of  displace- 
ment, being  in  fact  the  resultant  of  the  two  restoring  forces  which  its  two  components 
parallel  to  the  sides  of  the  rectangle  would  call  out. 


1126         POLARIZATION  AND  DOUBLE  REFRACTION. 

Thus  if  the  ellipse  in  the  annexed  figure  (Fig.  778)  represent  the 
vibrations  of  the  light  as  it  emerges  from  the  selenite,  and  CD, 
D  E  F  be  tangents  parallel  to  the  original  direction 

of  vibration,  the  perpendicular  distance  between 
these  tangents,  A  B,  is  the  component  vibration 
which  is  not  suppressed  when  the  analyser  is  so 
turned  that  all  the  light  would  be  suppressed  if 
the  selenite  were  removed.  By  rotating  the  analy- 
ser, we  shall  obtain  vibrations  of  various  amplitudes, 
corresponding  to  the  distances  between  parallel 
1  of  tangents  drawn  in  various  directions. 

For  a  certain  thickness  of  selenite  the  ellipse  may 
become  a  circle,  and  we  have  thus  what  is  called  circularly  polarized 
light,  which  is  characterized  by  the  property  that  rotation  of  the 
analyser  produces  no  change  of  intensity.  Circularly  polarized  light 
is  not  however  identical  with  ordinary  light;  for  the  interposition 
of  an  additional  thickness  of  selenite  converts  it  into  elliptically 
(or  in  a  particular  case  into  plane)  polarized  light  (§  1114). 

The  above  explanation  applies  to  homogeneous  light.  When  the 
incident  light  is  of  various  refrangibilities,  the  retardation  of  one 
component  upon  the  other  is  greatest  for  the  rays  of  shortest  wave- 
length. The  ellipses  are  accordingly  different  for  the  different  elemen- 
tary colours,  and  the  analyser  in  any  given  position  will  produce 
unequal  suppression  of  different  colours.  But  since  the  component 
which  is  suppressed  in  any  one  position  of  the  analyser,  is  the  com- 
ponent which  is  not  suppressed  when  the  analyser  is  turned  through 
a  right  angle,  the  light  yielded  in  the  former  case  plus  the  light 
yielded  in  the  latter  must  be  equal  to  the  whole  light  which  was 
incident  on  the  selenite.1  Hence  the  colours  exhibited  in  these  two 
positions  must  be  complementary. 

It  is  necessary  for  the  exhibition  of  colour  in  these  experiments 
that  the  plate  of  selenite  should  be  very  thin,  otherwise  the  retarda- 
tion of  one  component  vibration  as  compared  with  the  other  will  be 
greater  by  several  complete  periods  for  violet  than  for  red,  so  that 
the  ellipses  will  be  identical  for  several  different  colours,  and  the 
total  non-suppressed  light  will  be  sensibly  white  in  all  positions  of 
the  analyser. 

1  We  here  neglect  the  light  absorbed  and  scattered;  but  the  loss  of  this  does  not  sensibly 
affect  the  co'our  of  the  whole.  It  is  to  be  borne  in  mind  that  the  intensity  of  light  is  mea- 
sured by  the  square  of  the  amplitude,  and  is  therefore  the  simple  sum  of  the  intensities  of 
its  two  components  when  the  resolution  is  rectangular. 


EIXGS  AND   CROSS. 


1127 


Two  thick  plates  may  however  be  so  combined  as  to  produce  the 
effect  of  one  thin  plate.  For  example,  two  selenite  plates,  of  nearly 
equal  thickness,  may  be  laid  one  upon  the  other,  so  that  the  direc- 
tion of  greatest  elasticity  in  the  one  shall  be  parallel  to  that  of  least 
elasticity  in  the  other.  The  resultant  effect  in  this  case  will  be  that 
due  to  the  difference  of  their  thicknesses.  Two  plates  so  laid  are 
said  to  be  crossed. 

1109.  Colours  of  Plates  perpendicular  to  Axis. — A  different  class  of 


Fig.  779.— Rings  and  Cross. 

appearances  are  presented  when  a  plate,  cut  from  a  uniaxal  crystal 
by  sections  perpendicular  to  the  axis,  is  inserted  between  the  polar- 
izer and  the  analyser.  Instead  of  a  broad  sheet  of  uniform  colour, 
we  have  now  a  system  of  coloured  rings,  interrupted,  when  the 
analyser  is  in  one  of  the  two  critical  positions,  by  a  black  or  white 
cross,  as  at  A,  B  (Fig.  779). 

1110.  Explanation. — The  following  is  the  explanation  of  these  ap- 
pearances. Suppose,  for  simplicity,  that  the  analyser  is  a  plate  of 
tourmaline  held  close  to  the  eye.  Then  the  light  which  comes  to 
the  eye  from  the  nearest  point  of  the  plate  under 
examination  (the  foot  of  a  perpendicular  dropped 
upon  it  from  the  eye),  has  traversed  the  plate 
normally,  and  therefore  parallel  to  its  optic  axis. 
It  has  therefore  not  been  resolved  into  an  ordi- 
nary and  an  extraordinary  ray,  but  has  emerged 
from  the  plate  in  the  same  condition  in  which 
it  entered,  and  is  therefore  black,  gray,  or  white 
according  to  the  position  of  the  analyser,  just 
as  it  would  be  if  the  plate  were  removed.  But 
the  light  which  comes  obliquely  to  the  eye  from 
any  other  part  of  the  plate,  has  traversed  the  plate  obliquely,  and 
has  undergone  double  refraction.  Let  E  (Fig.  780)  be  the  position 


Fig.  780. 
Theory  of  Kings  and  Cross. 


1128  POLARIZATION   AND  DOUBLE   REFRACTION. 

of  the  eye,  E  O  a  perpendicular  on  the  plate,  P  a  point  on  the  cir- 
cumference of  a  circle  described  about  O  as  centre.  Then,  since 
E  0  is  parallel  to  the  axis  of  the  plate,  the  direction  of  vibration 
for  the  ordinary  ray  at  P  is  perpendicular  to  the  plane  E  O  P,  and 
is  tangential  to  the  circle.  The  direction  of  vibration  for  the  extra- 
ordinary ray  lies  in  the  plane  E  O  P,  is  nearly  perpendicular  to  E  O 
(or  to  the  axis),  if  the  angle  O  E  P  is  small,  and  deviates  more  from 
perpendicularity  to  the  axis  as  the  angle  O  E  P  increases.  Both  for 
this  reason,  and  also  on  account  of  the  greater  thickness  traversed, 
the  retardation  of  one  ray  upon  the  other  is  greater  as  P  is  taken 
further  from  O;  and  from  the  symmetry  of  the  circumstances,  it 
must  be  the  same  at  the  same  distance  from  0  all  round.  In  con- 
sequence of  this  retardation,  the  light  which  emerges  at  P  in  the  di- 
rection PE  is  elliptically  polarized;  and  by  the  agency  of  the  analyser 
it  is  accordingly  resolved  into  two  components,  one  of  which  is  sup- 
pressed. With  homogeneous  light,  rings  alternately  dark  and  bright 
would  thus  be  formed  at  distances  from  O  corresponding  to  retarda- 
tions of  0,  £,  1,  1J,  2,  2£,  .  .  .  complete  periods;  and  it  can  be  shown 
that  the  radii  of  these  rings  would  be  proportional  to  the  numbers  0, 
VI,  V2,  V3,  V4,  V5,  V6:  .  .  .  The  rings  are  larger  for  light  of  long 
than  of  short  wave-length;  and  the  coloured  rings  actually  exhibited 
when  white  light  is  employed,  are  produced  by  the  superposition  of 
all  the  systems  of  monochromatic  rings.  The  monochromatic  rings 
for  red  light  are  easily  seen  by  looking  at  the  actual  rings  through 
a  piece  of  red  glass. 

Let  O,  P,  Fig.  781,  be  the  same  points  which  were  denoted  by 
these  letters  in  Fig.  780,  and  let  A  B  be  the  direction  of  vibration  of 

the  light  incident  on  the  crystal  at  P. 
Draw  A  C,  D  B  parallel  to  O  P,  and 
complete    the    rectangle    A  C  B  D. 
Then  the  length  and  breadth  of  this 
rectangle  are  approximately  the  direc- 
tions of  vibration  of  the  two  com- 
ponents, one  of  which  loses  upon  the 
Fig.  78i. -Theory  of  Kings  and  Cross.        other  in  traversing  the  crystal.    The 
vibration  of  the  emergent  ray  is  re- 
presented by  an  ellipse  inscribed  in  the  rectangle  ACBD  (§  922, 
note  2) ;  and  when  the  loss  is  half  a  period,  this  ellipse  shrinks  into 
a  straight  line,  namely,  the  diagonal  C  D.     Through  C  and  D  draw 
lines  parallel  to  AB;   then  the  distance  between  these  parallels 


COLOUR  BY   ELLIPTIC   POLARIZATION.  1129 

represents  the  double  amplitude  of  the  vibration  which  is  trans- 
mitted when  there  has  been  a  retardation  of  half  a  period,  and  is 
greater  than  the  distance  between  the  tangents  in  the  same  direc- 
tion to  any  of  the  inscribed  ellipses.  A  retardation  of  another  half 
period  will  again  reduce  the  inscribed  ellipse  to  the  straight  line 
A  B,  as  at  first.  The  position  D  C  corresponds  to  the  brightest  and 
A  B  to  the  darkest  part  of  any  one  of  the  series  of  rings  for  a  given 
wave-length  of  light,  the  analyser  being  in  the  position  for  sup- 
pressing all  the  light  if  the  crystal  were  removed.  "When  the  analyser 
is  turned  into  the  position  at  right  angles  to  this,  A  B  corresponds 
to  the  brightest,  and  D  C  to  the  darkest  parts  of  the  rings.  It  is  to 
be  remembered  that  amount  of  retardation  depends  upon  distance 
from  the  centre  of  the  rings,  and  is  the  same  all  round.  The  two 
diagonals  of  our  rectangle  therefore  correspond  to  different  sizes  of 
rings. 

If  the  analyser  is  in  such  a  position  with  respect  to  the  point  P 
considered,  that  the  suppressed  vibration  is  parallel  to  one  of  the 
sides  of  the  rectangle  (in  other  v/ords,  if  O  P,  or  a  line  perpendi- 
cular to  0  P,  is  the  direction  of  suppression)  the  retardation  of  one 
component  upon  the  other  has  no  influence,  inasmuch  as  one  of  the 
two  components  is  completely  suppressed  and  the  other  is  completely 
transmitted.  There  are,  accordingly,  in  all  positions  of  the  analyser, 
a  pair  of  diameters,  coinciding  with  the  directions  of  suppression 
and  non-suppression,  which  are  alike  along  their  whole  length  and 
free  from  colour. 

Again  if  P  is  situated  at  B  or  at  90°  from  B,  the  corner  C  of  the 
rectangle  coincides  with  B  or  with  A,  and  the  rectangle,  with  all  its 
inscribed  ellipses,  shrinks  into  the  straight  line  AB.  The  two 
diameters  coincident  with  and  perpendicular  to  AB  are  therefore 
alike  along  their  whole  length  and  uncoloured. 

The  two  colourless  crosses  which  we  have  thus  accounted  for,  one 
of  them  turning  with  the  analyser  and  the  other  remaining  fixed 
with  the  polarizer,  are  easily  observed  when  the  analyser  is  not  near 
the  critical  positions.  In  the  critical  positions,  the  two  crosses  come 
into  coincidence;  and  these  are  also  the  positions  of  maximum  black- 
ness or  maximum  whiteness  for  the  two  crosses  considered  separ- 
ately. Hence  the  conspicuous  character  of  the  cross  in  either  of 
these  positions,  as  represented  at  A,  B,  Fig.  779.  As  the  analyser  is 
turned  away  from  these  positions,  the  cross  at  first  turns  after  it 
with  half  its  angular  velocity,  but  soon  breaks  up  into  rings,  some- 


1130  POLARIZATION  AND   DOUBLE   REFRACTION. 

what  in  the  manner  represented  at  C,  which  corresponds  to  a  posi- 
tion not  differing  much  from  A. 

1111.  Biaxal  Crystals.  —  Crystals  may  be  divided  optically  into 
three  classes: — 

1.  Those  in  which  there  is  no  distinction  of  different  directions,  as 
regards  optical  properties.     Such  crystals  are  said  to  be  optically 
isotropic. 

2.  Those  in  which  the  optical  properties  are  the  same  for  all  direc- 
tions equally  inclined  to  one  particular  direction  called  the  optic  axis, 
but  vary  according  to  this  inclination.     Such  crystals  are  called 
uniaxal. 

3.  All  remaining  crystals  (excluding  compound  and  irregular  for- 
mations) belong  to  the  class  called  biaxal.     In  any  homogeneous 
elastic  solid,  there  are  three  cardinal  directions  called  axes  of  elasti- 
city, possessing  the  same  distinctive  properties  which  belong  to  the 
two  principal  planes  of  vibration  in  Blackburn's  pendulum  (§  924); 
that  is  to  say,  if  any  small  portion  of  the  solid  be  distorted  by  for- 
cibly displacing  one  of  its  particles  in  one  of  these  cardinal  directions, 
the  forces  of  elasticity  thus  evoked  te/id  to  Urge  the  particle  directly 
back;  whereas  displacement  in  any  other  direction  calls  out  forces 
whose  resultant  is  generally  oblique  to  the  direction  of  displacement, 
so  that  when  the  particle  is  released  it  does  not  fly  back  through  the 
position  of  equilibrium,  but  passes  on  one  side  of  it,  just  as  the  bob 
of  Blackburn's  pendulum  generally  passes  beside  and  not  through 
the  lowest  point  which  it  can  reach. 

In  biaxal  crystals,  the  resistances  to  displacement  in  the  three 
cardinal  directions  are  all  unequal;  and  this  is  true  not  only  for  the 
crystalline  substance  itself,  but  also  for  the  luminiferous  aether  which 
pervades  it,  and  is  influenced  by  it.1  The  construction  given  by 
Fresnel  for  the  wave-surface  in  any  crystal  is  as  follows: — First 
take  an  ellipsoid,  having  its  axis  parallel  to  the  three  cardinal  .direc- 
tions, and  of  lengths  depending  on  the  particular  crystalline  sub- 
stance considered.  Then  let  any  plane  sections  (which  will  of  course 
be  ellipses)  be  made  through  the  centre  of  this  ellipsoid,  let  normals 
to  them  be  drawn  through  the  centre,  -and  on  each  normal  let  points 
be  taken  at  distances  from  the  centre  equal  to  the  greatest  and  least 
radii  of  the  corresponding  section.  The  locus  of  these  points  is  the 
complete  wave-surface,  which  consists  of  two  sheets  cutting  one 

1  The  cardinal  directions  are  however  believed  not  to  be  the  same  for  the  aether  as  for 
the  material  of  the  crystal. 


BIAXAL   CRYSTALS.  1131 

another  at  four  points.  These  four  points  of  intersection  are  situated 
upon  the  normals  to  the  two  circular  sections  of  the  ellipsoid,  and 
the  two  optic  axes,  from  which  biaxal  crystals  derive  their  name, 
are  closely  related  to  these  two  circular  sections.  The  optic  axes 
are  the  directions,  of  single  wave-velocity,  and  the  normals  to  the 
two  circular  sections  are  the  directions  of  single  ray-velocity.  The 
direction  of  advance  of  a  wave  is  always  regarded  as  normal  to  the 
front  of  the  wave,  whereas  the  direction  of  a  ray  (denned  by  the 
condition  of  traversing  two  apertures  placed  in  its  path)  always 
passes  through  the  centre  of  the  wave-surface,  and  is  not  in  general 
normal  to  the  front.  Both  these  pairs  of  directions  of  single  velo- 
city are  in  the  plane  which  contains  the  greatest  and  least  axes  of 
the  ellipsoid. 

When  two  axes  of  the  ellipsoid  are  equal,  it  becomes  a  spheroid, 
and  the  crystal  is  uniaxal.  When  all  three  axes  are  equal,  it  becomes 
a  sphere,  and  the  crystal  is  isotropic. 

Experiment  has  shown  that  biaxal  crystals  expand  with  heat 
unequally  in  three  cardinal  directions,  so  that  in  fact  a  spherical 
piece  of  such  a  crystal  is  changed  into  an  ellipsoid1  when  its  tem- 
perature is  raised  or  lowered.  A  spherical  piece  of  a  uniaxal  crystal 
in  the  same  circumstances  changes  into  a  spheroid;  and  a  spherical 
piece  of  an  isotropic  crystal  remains  a  sphere. 

It  is  generally  possible  to  determine  to  which  of  the  three  classes 
a  crystal  belongs,  from  a  mere  inspection  of  its  shape  as  it  occurs  in 
nature.  Isotropic  crystals  are  sometimes  said  to  be  symmetrical 
about  a  point,  uniaxal  crystals  about  a  line,  biaxal  crystals  about 
neither.  The  following  statement  is  rather  more  precise:— 

If  there"  is  one  and  only  one  line  about  which  if  the  crystal  be 
rotated  through  90°  or  else  through  120°  the  crystalline  form  remains 
in  its  original  position,  the  crystal  is  uniaxal,  having  that  line  for 
the  axis.  If  there  is  more  than  one  such  line,  the  crystal  is  isotropic, 
while,  if  there  is  no  such  line,  it  is  biaxal.  Even  in  the  last  case,  if 
there  exist  a  plane  of  crystalline  symmetry,  such  that  one  half  of 
the.  crystal  is  the  reflected  image  of  the  other  half  with  respect  to 
this  plane,  it  is  also  a  plane  of  optical  symmetry,  and  one  of  the  three 
cardinal  directions  for  the  aether  is  perpendicular  to  it.2 

1  This  fact  furnishes  the  best  possible  definition  of  an  ellipsoid  for  persons  unacquainted 
with  solid  geometry. 

*  The  optic  axes  either  lie  in  the  plane  of  symmetry,  or  lie  in  a  perpendicular  plane  and 
are  equally  inclined  to  the  plane  of  symmetry. 

For  the  precise  statement  here  given,  the  Editor  is  indebted  to  Professor  Stokes. 


1132        POLARIZATION  AND  DOUBLE  REFRACTION. 

Glass,  when  in  a  strained  condition,  ceases  to  be  isotropic,  and  if 
inserted  between  a  polarizer  and  an  analyser,  exhibits  coloured 
streaks  or  spots,  which  afford  an  indication  of  the  distribution  of 
strain  through  its  substance.  The  experiment  is  shown  sometimes 
with  unannealed  glass,  which  is  in  a  condition  of  permanent  strain, 
sometimes  with  a  piece  of  ordinary  glass  which  can  be  subjected  to 
force  at  pleasure  by  turning  a  screw.  Any  very  small  portion  of  a 
piece  of  strained  glass  has  the  optical  properties  of  a  crystal,  but 
different  portions  have  different  properties,  and  hence  the  glass  as  a 
whole  does  not  behave  like  one  crystal. 

The  production  of  colour  by  interposition  between  a  polarizer  and 
an  analyser,  is  by  far  the  most  delicate  test  of  double  refraction. 
Many  organic  bodies  (for  example,  grains  of  starch)  are  thus  found 
to  be  doubly  refracting;  and  microscopists  often  avail  themselves  of 
this  means  of  detecting  diversities  of  structure  in  the  objects  which 
they  examine. 

1112.  Rotation  of  Plane  of  Polarization. — When  a  plate  of  quartz 
(rock-crystal),  even  of  considerable  thickness,  cut  perpendicular  to 
the  axis,  is  interposed  between  the  polarizer  and  analyser,  colour  is 
exhibited,  the  tints  changing  as  the  analyser  is  rotated;  and  similar 
effects  of  colour  are  produced  by  employing,  instead  of  quartz,  a 
solution  of  sugar,  inclosed  in  a  tube  with  plane  glass  ends. 

If  homogeneous  light  is  employed,  it  is  found  that  if  the  analyser 
is  first  adjusted  to  produce  extinction  of  the  polarized  light,  and  the 
quartz  or  saccharine  solution  is  then  introduced,  there  is  a  partial 
restoration  of  light.  On  rotating  the  analyser  through  a  certain 
angle,  there  is  again  complete  extinction  of  the  light;  and  on  com- 
paring different  plates  of  quartz,  it  will  be  found  that  the  angle 
through  which  the  analyser  must  be  rotated  is  proportional  to  the 
thickness  of  the  plate.  In  the  case  of  solutions  of  sugar,  the  angle 
is  proportional  jointly  to  the  length  of  the  tube  and  the  strength  of 
the  solution. 

The  action  thus  exerted  by  quartz  or  sugar  is  called  rotation  of 
the  plane  of  polarization,  a  name  which  precisely  expresses  the 
observed  phenomena.  In  the  case  of  ordinary  quartZj  and  solutions 
of  sugar-candy,  it  is  necessary  to  rotate  the  analyser  in  the  direc- 
tion of  watch-hands  as  seen  by  the  observer,  and  the  rotation  of 
the  plane  of  polarization  is  said  to  be  right-handed.  In  the  case 
of  what  is  called  left-handed  quartz,  and  of  solutions  of  non-crystal - 
lizable  sugar,  the  rotation  of  the  plane  of  polarization  is  in  the 


ROTATION   OF  PLANE   OF  POLARIZATION.  1133 

opposite  direction,  and  the  observer  must  rotate  the  analyser  against 
watch-hands. 

The  amount  of  rotation  is  different  for  the  different  elementary 
colours,  and  has  been  found  to  be  inversely  as  the  square  of  the 
wave-length.  Hence  the  production  of  colour. 

1113.  Magneto-optic  Rotation. — Faraday  made  the  remarkable  dis- 
covery that  the  plane  of  polarization  can  be  rotated  in  certain 
circumstances  by  the  action  of  magnetism.  Let  a  long  rectangular 
piece  of  "  heavy-glass  "  (silico-borate  of  lead)  be  placed  longitudinally 
between  the  poles  of  the  powerful  electro-magnet  represented  in 
Fig.  445  (page  683),  which  is  for  this  purpose  made  hollow  in  its 
axis,  so  that  an  observer  can  see  through  it  from  end  to  end.  Let  a 
Nicol's  prism  be  fitted  into  one  end  of  the  magnet,  to  serve  as  polar- 
izer, and  another  into  the  other  end  to  serve  as  analyser,  and  let  one 
of  them  be  turned  till  the  light  is  extinguished.  Then,  as  long  as 
no  current  is  passed  round  the  electro-magnet,  the  interposition  of 
the  heavy-glass  will  produce  no  effect;  but  the  passing  of  a  current 
while  the  heavy-glass  is  in  its  place  between  the  poles,  produces 
rotation  of  the  plane  of  polarization  in  the  same  direction  as  that  in 
which  the  current  circulates.  The  amount  of  rotation  is  directly  as 
the  strength  of  current,  and  directly  as  the  length  of  heavy-glass 
traversed  by  the  light.  Flint-glass  gives  about  half  the  effect  of 
heavy-glass,  and  all  transparent  solids  and  liquids  exhibit  an  effect 
of  the  same  kind  in  a  more  or  less  marked  degree. 

A  steel  magnet,  if  extremely  powerful,  may  be  used  instead  of  an 
electro-magnet;  and  in  all  cases,  to  give  the  strongest  effect,  the 
lines  of  magnetic  force  should  coincide  with  the  direction  of  the 
transmitted  ray. 

Faraday  regarded  these  phenomena  as  proving  the  direct  action 
of  magnetism  upon  light;  but  it  is  now  more  commonly  believed 
that  the  direct  effect  of  the  magnetism  is  to  put  the  particles  of  the 
transparent  body  in  a  peculiar  state  of  strain,  to  which  the  observed 
optical  effect  is  due. 

In  -every  case  tried  by  Faraday,  the  direction  of  the  rotation  was 
the  same  as  the  direction  in  which  the  current  circulated;  but  cer- 
tain substances1  have  since  been  found  which  give  rotation  against 
the  current.  The  law  for  the  relative  amounts  of  rotation  of  differ- 
ent colours  is  approximately  the  same  as  in  the  case  of  quartz. 

1  One  such  substance  is  a  solution  of  Fe'Cl3  (old  notation)  in  methylic  (not  methylated) 
alcohol 

72 


1134  POLARIZATION   AND   DOUBLE   REFRACTION. 

The  direction  of  rotation  is  with  watch-hands  as  seen  from  one  end 
of  the  arrangement,  and  against  watch-hands  as  seen  from  the  other; 
so  that  the  same  piece  of  glass,  in  the  same  circumstances,  behaves 
like  right-handed  quartz  to  light  entering  it  at  one  end,  and  like 
left-handed  quartz  to  light  entering  it  at  the  other. 

The  rotatory  power  of  quartz  and  sugar  appears  to  depend  upon  a 
certain  unsymmetrical  arrangement  of  their  molecules,  an  arrange- 
ment somewhat  analogous  to  the  thread  of  a  screw;  right-handed 
and  left-handed  screws  representing  the  two  opposite  rotatory 
powers.  It  is  worthy  of  note  that  the  two  kinds  of  quartz  crystallize 
in  different  forms,  each  of  which  is  unsymmetrical,  one  being  like 
the  image  of  the  other  as  seen  in  a  looking-glass.  Pasteur  has  con- 
ducted extremely  interesting  researches  into  the  relations  existing 
between  substances  which,  while  in  other  respects  identical  or  nearly 
identical,  differ  as  regards  their  power  of  producing  rotation.  For 
the  results  we  must  refer  to  treatises  on  chemistry. 

Dr.  Kerr  has  recently  obtained  rotation  of  the  plane  of  polariza- 
tion by  reflection  from  intensely  magnetized  iron.  In  some  of  the 
experiments  the  direction  of  magnetization  was  normal,  and  in  others 
parallel  to  the  reflecting  surface. 

1114.  Circular  Polarization.  Fresnel's  Rhomb. — We  have  explained 
in  §  1108  the  process  by  which  elliptic  polarization  is  brought  about, 
when  plane-polarized  light  is  transmitted  through  a  thin  plate  of 
selenite.  To  obtain  circular  polarization  (which  is  merely  a  case  of 
elliptic),  the  plate  must  be  of  such  thickness  as  to  retard  one  com- 
ponent more  than  the  other  by  a  quarter  of  a  wave-length,  and  must 
be  laid  so  that  the  directions  of  the  two  component  vibrations  make 
angles  of  45°  with  the  plane  of  polarization.  Plates  specially  pre- 
pared for  this  purpose  are  in  general  use,  and  are  called  quarter- 
wave  plates.  They  are  usually  of  mica,  which  differs  but  little  in  its 
properties  from  selenite.  It  is  impossible,  however,  in  this  way  to 
obtain  complete  circular  polarization  of  ordinary  white  light,  since 
different  thicknesses  are  required  for  light  of  different  wave-lengths, 
the  thickness  which  is  appropriate  for  violet  being  too  small  for 
red. 

Fresnel  discovered  that  plane-polarized  light  is  elliptically  polar- 
ized by  total  internal  reflection  in  glass,  whenever  the  plane  of 
polarization  of  the  incident  light  is  inclined  to  the  plane  of  inci- 
dence. The  rectilinear  vibrations  of  the  incident  light  are  in  fact 
resolved  into  two  components,  one  of  them  in,  and  the  other  per- 


CIRCULAR  POLARIZATION. 


1135 


pendicular  to,  the  plane  of  incidence;  and  one  of  these  is  retarded 
with  respect  to  the  other  in  the  act  of  reflection,  by  an  amount  de- 
pending on  the  angle  of  incidence.  He  determined  the  magnitude 
of  this  angle  for  which  the  retardation  is  precisely  £  of  a  wave-length ; 
and  constructed  a  rhomb,  or  oblique  parallelepiped 
of  glass  (Fig.  782),  in  which  a  ray,  entering  normally 
at  one  end,  undergoes  two  successive  reflections  at 
this  angle  (about  55°),  the  plane  of  reflection  being 
the  same  in  both.  The  total  retardation  of  one 
component  on  the  other  is  thus  £  of  a  wave-length; 
and  if  the  rhomb  is  in  such  a  position  that  the 
plane  in  which  the  two  reflections  take  place  is  at 
an  angle  of  45°  to  the  plane  of  polarization  of  the 
incident  light,  the  emergent  light  is  circularly 
polarized.  The  effect  does  not  vary  much  with  the 
wave-length,  and  sensibly  white  circularly  polarized 
light  can  accordingly  be  obtained  by  this  method. 

When  circularly  polarized  light  is  transmitted 
through  a  Fresnel's  rhomb,  or  through  a  quarter- 
wave  plate,  it  becomes  plane-polarized,  and  we 
have  thus  a  simple  mode  of  distinguishing  circularly  polarized 
light  from  common  light;  for  the  latter  does  not  become  polarized 
when  thus  treated.  Two  quarter-wave  plates,  or  two  Fresnel's 
rhombs,  may  be  combined  either  so  as  to  assist  or  to  oppose  one 
another.  By  the  former  arrangement,  which  is  represented  in  Fig. 
782,  we  can  convert  plane-polarized  light  into  light  polarized  in  a 
perpendicular  plane,  the  final  result  being  therefore  the  same  as  if 
the  plane  of  polarization  had  been  rotated  through  90°.  The  several 
steps  of  the  process  are  illustrated  by  the  five  diagrams  of  Fig.  783, 


Fig.  782. 
Two  Freanel's  Rhombs. 


Fig.  783.— Form  of  Vibration  in  traversing  the  Rhombs. 

which  represent  the  vibrations  of  the  five  portions  AC,  CD,  T)d,  dc,ca 
of  the  ray  which  traverses  the  two  rhombs  in  the  preceding  figure. 
The  sides  of  the  square  are  parallel  to  the  directions  of  resolution; 
the  initial  direction  of  vibration  is  one  diagonal  of  the  square,  and 


1136        POLARIZATION  AND  DOUBLE  REFRACTION. 

the  final  direction  is  the  other  diagonal;  a  gain  or  loss  of  half  a  com- 
plete vibration  on  the  part  of  either  component  being  just  sufficient 
to  effect  this  change. 

1115.  Direction  of  Vibration  of  Plane-polarized  Light. — The  plane  of 
polarization  of  plane-polarized  light  may  be  defined  as  the  plane  in 
which  it  is  most  copiously  reflected.    It  is  perpendicular  to  the  plane 
in  which  the  light  refuses  to  be  reflected  (at  the  polarizing  angle); 
and  is  identical  with  the  original  plane  of  reflection,  if  the  polariza- 
tion was  produced  by  reflection.     This  definition  is  somewhat  arbi- 
trary, but  has  been  adopted  by  universal  consent. 

When  light  is  polarized  by  the  double  refraction  of  Iceland-spar, 
or  of  any  other  uniaxal  crystal,  it  is  found  that  the  plane  of  polariza- 
tion of  the  .ordinary  ray  is  the  plane  which  contains  the  axis  of  the 
crystal.  But  the  distinctive  properties  of  the  ordinary  ray  are  most 
naturally  explained  by  supposing  that  its  vibrations  are  perpendi- 
cular to  the  axis.  Hence  we  conclude  that  the  direction  of  vibration 
in  plane-polarized  light  is  normal  to  the  so-called  plane  of  polariza- 
tion, and  therefore  that,  in  polarization  by  reflection,  the  vibrations 
of  the  reflected  light  are  parallel  to  the  reflecting  surface. 

This  is  Fresnel's  doctrine.  MacCullagh,  however,  reversed  this 
hypothesis,  and  maintained  that  the  direction  of  vibration  is  in  the 
plane  of  polarization.  Both  theories  have  been  ably  expounded;  but 
Stokes  contrived  a  crucial  experiment  in  diffraction,  which  confirmed 
Fresnel's  view;1  and  in  his  classical  paper  on  "Change  of  Refrangi- 
bility,"  he  has  deduced  the  same  conclusion  from  a  consideration  of 
the  phenomena  of  the  polarization  of  light  by  reflection  from  exces- 
sively fine  particles  of  solid  matter  in  suspension  in  a  liquid.2 

1116.  Vibrations  of  Ordinary  Light. — Ordinary  light  agrees  with 
circularly  polarized  light  in  always  yielding  two  beams  of  equal 
intensity  when  subjected  to  double  refraction;  but  it  differs  from 
circularly  polarized  light  in  not  becoming  plane-polarized  by  trans- 
mission through  a  Fresnel's  rhomb  or  a  quarter- wave  plate.     What, 
then,  can  be  the  form  of  vibration  for  common  light  ?    It  is  probably 
very  irregular,  consisting  of  ellipses  of  various  sizes,  positions,  and 
forms  (including  circles  and  straight  lines),  rapidly  succeeding  one 
another.     By  this  irregularity  we  can  account   for  the  fact  that 
beams  of  light  from  different  sources  (even  from  different  points  of 
the  same  flame,  or  from  different  parts  of  the  sun's  disc),  cannot,  by 

1  Cambridge  Transactions.     1850. 

2  Philosophical  Transactions,  1852;  pp.  530,  531. 


VIBRATIONS   OF   ORDINARY   LIGHT.  1137 

any  treatment  whatever,  be  made  to  exhibit  the  phenomena  of 
mutual  interference;  and  for  the  additional  fact  that  the  two  rect- 
angular components  into  which  a  beam  of  common  light  is  resolved 
by  double  refraction,  cannot  be  made  to  interfere,  even  if  their 
planes  of  polarization  are  brought  into  coincidence  by  one  of  the 
methods  of  rotation  above  described. 

Certain  phenomena  of  interference  show  that  a  few  hundred  con- 
secutive vibrations  of  common  light  may  be  regarded  as  similar;  but 
as  the  number  of  vibrations  in  a  second  is  about  500  millions  of 
millions,  there  is  ample  room  for  excessive  diversity  during  the  time 
that  one  impression  remains  upon  the  retina. 

1117.  Polarization  of  Radiant  Heat. — The  fundamental  identity  of 
radiant  heat  and  light  is  confirmed  by  thermal  experiments  on 
polarization.  Such  experiments  were  first  successfully  performed  by 
Forbes  in  1834,  shortly  after  Melloni's  invention  of  the  thermo- 
multiplier.  He  first  proved  the  polarization  of  heat  by  tourmaline; 
next  by  transmission  through  a  bundle  of  very  thin  mica  plates, 
inclined  to  the  transmitted  rays;  and  afterwards  by  reflection  from 
the  multiplied  surfaces  of  a  pile  of  thin  mica  plates  placed  at  the 
polarizing  angle.  He  next  succeeded  in  showing  that  polarized  heat, 
even  when  quite  obscure,  is  subject  to  the  same  modifications  which 
doubly  refracting  crystallized  bodies  impress  upon  light,  by  suffering 
a  beam  of  heat,  after  being  polarized  by  transmission,  to  pass  through 
an  interposed  plate  of  mica,  serving  the  purpose  of  the  plate  of  selenite 
in  the  experiment  of  §  1107,  the  heat  traversing  a  second  mica  bundle 
before  it  was  received  on  the  thermo-pile.  As  the  interposed  plate 
was  turned  round  in  its  own  plane,  the  amount  of  heat  shown  by  the 
galvanometer  was  found  to  fluctuate  just  as  the  amount  of  light 
received  by  the  eye  under  similar  circumstances  would  have  done. 
He  also  succeeded  in  producing  circular  polarization  of  heat  by  a 
Fresnel's  rhomb  of  rock-salt.  These  results  have  since  been  fully 
confirmed  by  the  experiments  of  other  observers. 


EXAMPLES  IN  ACOUSTICS. 


PERIOD,  WAVE-LENGTH,  AND  VELOCITY. 

1.  If  an  undulation  travels  at  the  rate  of  100  ft.  per  second,  and  the  wave- 
length is  2  ft.,  find  the  period  of  vibration  of  a  particle,  and  the  number  of  vibra- 
tions which  a  particle  makes  per  second. 

2.  It  is  observed  that  waves  pass  a  given  point  once  in  every  5  seconds,  and 
that  the  distance  from  crest  to  crest  is  20  ft.     Find  the  velocity  of  the  waves  in 
feet  per  second. 

3.  The  lowest  and  highest  notes  of  the  normal  human  voice  have  about  80  and 
800  vibrations  respectively  per  second.     Find  their  wave-lengths  when  the  velo- 
city of  sound  is  1100  ft.  per  second. 

4.  Find  their  wave-lengths  in  water  in  whicli  the  velocity  of  sound  is  4900  feet 
per  second. 

5.  Find  the  wave-length  of  a  note  of  500  vibrations  per  second  in  steel  in 
which  the  velocity  of  propagation  is  15,000  ft.  per  second. 

PITCH  AND  MUSICAL  INTERVALS. 

6.  Show  that  a  "fifth"  added  to  a  "fourth"  makes  an  octave. 

7.  Calling  the  successive  notes  of  the  gamut  Doi,  E,CI,  Mil,  Fa1?  Soli,  Lai,  Sii, 
Do*,  show  that  the  interval  from  Soli  to  Ee2  is  a  true  "fifth." 

8.  Find  the  first  5  harmonics  of  Doi. 

9.  A  siren  of  15  holes  makes  2188  revolutions  in  a  minute  when  in  unison 
with  a  certain  tuning-fork.    Find  the  number  of  vibrations  per  second  made  by 
the  fork. 

10.  A  siren  of  15  holes  makes  440  revolutions  in  a  quarter  of  a  minute  when 
in  unison  with  a  certain  pipe.    Find  the  note  of  the  pipe  (in  vibrations  per 
second). 

REFLECTION  OF  SOUND,  AND  TONES  OF  PIPES. 

11.  Find  the  distance  of  an  obstacle  which  sends  back  the  echo  of  a  sound  to 
the  source,  in  H  seconds,  when  the  velocity  of  sound  is  1100  ft.  per  second. 

12.  A  well  is  210  ft.  deep  to  the  surface  of  the  water.     What  time  will  elapse 
between  producing  a  sound  at  its  mouth  and  hearing  the  echo? 

13.  What  is  the  wave-length  of  the  fundamental  note  of  an  open  organ-pipe 
16  ft.  long;  and  what  are  the  wave-lengths  of  its  first  two  overtones?     Find  also 
their  vibration-numbers  per  second. 

14.  What  is  the  wave-length  of  the  fundamental  tone  of  a  stopped  organ  pipe 


EXAMPLES   IN   ACOUSTICS.  1139 

5  ft.  long;  and  what  are  the  wave-lengths  of  its  first  two  overtones?    Find  also 
their  vibration-numbers  per  second. 

15.  What  should  be  the  length  of  a  tube  stopped  at  one  end  that  it  may 
resound  to  the  note  of  a  tuning-fork  which  makes  520  vibrations  per  second;  and 
what  should  be  the  length  of  a  tube  open  at  both  ends  that  it  may  resound  to  the 
same  fork.     [The  tubes  are  supposed  narrow,  and  the  smallest  length  that  will 
suffice  is  intended.] 

16.  Would  tubes  twice  as  long  as  those  found  in  last  question  resound  to  the 
fork?    Would  tubes  three  times  as  long? 

BEATS. 

17.  One  fork  makes  256  vibrations  per  second,  and  another  makes  260.    How 
many  beats  will  they  give  in  a  second  when  sounding  together1? 

18.  Two  sounds,  each  consisting  of  a  fundamental  tone  with  its  first  two  har- 
monics, reach  the  ear  together.     One  of  the  fundamental  tones  has  300  and  the 
other  302  vibrations  per  second.     How  many  beats  per  second  are  due  to  the 
fundamental  tones,  how  many  to  the  first  harmonics,  and   how  many  to  the 
second  harmonics? 

19.  A  note  of  225  vibrations  per  second,  and  another  of  336  vibrations  per 
second,  are  sounded  together.     Each  of  the  two  notes  contains  the  first  two  har- 
monics of  the  fundamental.     Show  that  two  of  the  harmonics  will  yield  beats  at 
the  rate  of  3  per  second. 

VELOCITY  OF  SOUND  IN  GASES. 

20.  If  the  velocity  of  sound  in  air  at  0°  C.  is  33,240  cm.  per  second,  find  its 
velocity  in  air  at  10°  C.,  and  in  air  at  100°  C. 

21.  If  the  velocity  of  sound  in  air  at  0°  C.  is  1090  ft.  per  second,  what  is  the 
velocity  in  air  at  10°? 

22.  Show  that  the  difference  of  velocity  for  1°  of  difference  of  temperature  in 
the  Fahrenheit  scale  is  about  1  ft.  per  second. 

23.  If  the  wave-length  of  a  certain  note  be  1  metre  in  air  at  0°,  what  is  it  in 
air  at  10°? 

24.  The  density  of  hydrogen  being  '06926  of  that  of  air  at  the  same  pressure 
and  temperature,  find  the  velocity  of  sound  in  hydrogen  at  a  temperature  at  which 
the  velocity  in  air  is  1100  ft.  per  second. 

25.  The  quotient  of  pressure  (in  dynes  per  sq.  cm.)  by  density  (in  gm.  per  cubic 
cm.)  for  nitrogen  at  0°  C.  is  807  million.   Compute  (in  cm.  per  second)  the  velocity 
of  sound  in  nitrogen  at  this  temperature. 

26.  If  a  pipe  gives  a  note  of  512  vibrations  per  second  in  air,  what  note  will 
it  give  in  hydrogen  ? 

27.  A  pipe  gives  a  note  of  100  vibrations  per  second  at  the  temperature  10°  C. 
What  must  be  the  temperature  of  the  air  that  the  same  pipe  may  yield  a  note 
higher  by  a  major  fifth  1 

VIBRATIONS  OF  STRINGS. 

28.  Find,  in  cm.  per  second,  the  velocity  with  which  pulses  travel  along  a 
string  whose  mass  per  cm.  of  length  is  '005  gm.,  when  stretched  with  a  force  of 
7  million  dynes. 


1140  EXAMPLES   IN    OPTICS. 

29.  If  the  length  of  the  string  in  last  question  be  33  cm.,  find  the  number  of 
vibrations  that  it  makes  per  second  when  vibrating  in  its  fundamental  mode;  also 
the  numbers  corresponding  to  its  first  two  overtones. 

30.  The  A  string  of  a  violin  is  33  cm.  long,  has  a  mass  of  '0065  gm.  per  cm., 
and  makes  440  vibrations  per  second.     Find  the  stretching  force  in  dynes. 

31.  The  E  string  of  a  violin  is  33  cm.  long,  has  a  mass  of  '004  gm.  per  cm.,  and 
makes  660  vibrations  per  second.     Find  the  stretching  force  in  dynes. 

32.  Two  strings  of  the  same  length  and  section  are  formed  of  materials  whose 
specific  grayities  are  respectively  d  and  d'.   Each  of  these  strings  is  stretched  with 
a  weight  equal   to  1000  times  its  own  weight.     What  is  the  musical  interval 
between  the  notes  which  they  will  yield? 

33.  The  specific  gravity  of  platinum  being  taken  as  22,  and  that  of  iron  as  7'8> 
what  must  be  the  ratio  of  the  lengths  of  two  wires,  one  of  platinum  and  the  other 
of  iron,  both  of  the  same  section,  that  they  may  vibrate  in  unison  when  stretched 
with  equal  forces? 

LONGITUDINAL  VIBRATIONS  OF  EODS. 

34.  If  sound  travels  along  fir  in  the  direction  of  the  fibres  at  the  rate  of  15,000  ft. 
per  second,  what  must  be  the  length  of  a  fir  rod  that,  when  vibrating  longi- 
tudinally in  its  fundamental  mode,  it  may  emit  a  note  of  750  vibrations  per 
second  ? 

35.  A  rod  8  ft.  long,  vibrating  longitudinally  in  its  fundamental  mode,  gives  a 
note  of  800  vibrations  per  second.     Find  the  velocity  with  which  pulses  are  pro- 
pagated along  it. 


EXAMPLES   IN   OPTICS. 


PHOTOMETRY,  SHADOWS,  AND  PLANE  MIRRORS. 

36.  A  lamp  and  a  taper  are  at  a  distance  of  4'15  m.  from  each  other;  and  it 
is  known  that  their  illuminating  powers  are  as  6  to  1.   At  what  distance  from  the 
lamp,  in  the  straight  line  joining  the  flames,  must  a  screen  be  placed  that  it  may 
be  equally  illuminated  by  them  both? 

37.  Two  parallel  plane  mirrors  face  each  other  at  a  distance  of  3  ft.,  and  a 
small  object  is  placed  between  them  at  a  distance  of  1  ft.  from  the  first  mirror, 
and  therefore  of  2  ft.  from  the  second.     Calculate  the  distances,  from  the  first 
mirror,  of  the  three  nearest  images  which  are  seen  in  it;  and  make  a  similar  cal- 
culation for  the  second  mirror. 

38.  Show  that  a  person  standing  upright  in  front  of  a  vertical  plane  mirror 
will  just  be  able  to  see  his  feet  in  it,  if  the  top  of  the  mirror  is  on  a  level  with  his 
eyes,  and  its  height  from  top  to  bottom  is  half  the  height  of  his  eyes  above  his 
feet. 

39.  A  square  plane  mirror  hangs  exactly  in  the  centre  of  one  of  the  walls  of  a 
cubical  room.     What  must  be  the  size  of  the  mirror  that  an  observer  with  his 


EXAMPLES   IN   OPTICS.  1141 

eye  exactly  in  the  centre  of  the  room  may  just  be  able  to  see  the  whole  of  the 
opposite  wall  reflected  in  it  except  the  part  concealed  by  his  body? 

40.  Two  plane  mirrors  contain  an  angle  of  160°,  and  form  images  of  a  small 
object  between  them.     Show  that  if  the  object  be  within  20°  of  either  mirror 
there  will  be  three  images ;  and  that  if  it  be  more  than  20°  from  both,  there  will 
be  only  two. 

41.  Show  that  when  the  sun  is  shining  obliquely  on  a  plane  mirror,  an  object 
directly  in  front  of  the  mirror  may  give  two  shadows,  besides  the  direct  shadow. 

42.  A  person  standing  beside  a  river  near  a  bridge  observes  that  the  inverted 
image  of  the  concavity  of  the  arch  receives  his  shadow  exactly  as  a  real  inverted 
arch  would  do  if  it  were  in  the  place  where  the  image  appears  to  be.     Explain 
this. 

43.  If  a  globe  be  placed  upon  a  table,  show  that  the  breadth  of  the  elliptic 
shadow  cast  by  a  candle  (considered  as  a  luminous  point)  will  be  independent  of 
the  position  of  the  globe. 

44.  What  is  the  length  of  the  cone  of  the  umbra  thrown  by  the  earth?  and 
what  is  the  diameter  of  a  cross  section  of  it  made  at  a  distance  equal  to  that  of 
the  moon? 

The  radius  of  the  sun  is  112  radii  of  the  earth;  the  distance  of  the  moon  from 
the  earth  is  60  radii  of  the  earth;  and  the  distance  of  the  sun  from  the  earth  is 
24,000  radii  of  the  earth.  Atmospheric  refraction  is  to  be  neglected. 

45.  The  stem  of  a  siren  carries  a  plane  mirror,  thin,  polished  on  both  sides,  and 
parallel  to  the  axis  of  the  stem.     The  siren  gives  a  note  of  345  vibrations  per 
second.     The  revolving  plate  has  15  holes.     A  fixed  source  of  light  sends  to  the 
mirror  a  horizontal  pencil  of  parallel  rays.     What  space  is  traversed  in  a  second 
by  a  point  of  the  reflected  pencil  at  a  distance  of  4  metres  from  the  axis  of  the 
siren  ?     This  axis  is  supposed  vertical. 

SPHERICAL  MIRRORS. 

46.  Find  the  focal  length  of  a  concave  mirror  whose  radius  of  curvature  ia 
2  ft.,  and  find  the  position  of  the  image  (a)  of  a  point  15  in.  in  front  of  the  mirror; 
(6)  of  a  point  10  ft.  in  front  of  the  mirror;  (c)  of  a  point  9  in.  in  front  of  the 
mirror ;  (d)  of  a  point  1  in.  in  front  of  the  mirror. 

47.  Calling  the  diameter  of  the  object  unity,  find  the  diameters  of  the  image 
in  the  four  preceding  cases. 

48.  The  flame  of  a  candle  is  placed  on  the  axis  of  a  concave  spherical  mirror 
at  the  distance  of  154  cm.,  and  its  image  is  formed  at  the  distance  of  45  cm.    What 
is  the  radius  of  curvature  of  the  mirror? 

49.  On  the  axis  of  a  concave  spherical  mirror  of  1  m.  radius,  an  object  9  cm. 
high  is  placed  at  a  distance  of  2  m.     Find  the  size  and  position  of  the  image. 

50.  What  is  the  size  of  the  circular  image  of  the  sun  which  is  formed  at  the 
principal  focus  of  a  mirror  of  20  m.  radius?     The  apparent  diameter  of  the  sun 
is  30'. 

51.  In  front  of  a  concave  spherical  mirror  of  2  metres'  radius  is  placed  a  con- 
cave luminous  arrow,  1  decimetre  long,  perpendicular  to  the  principal  axis,  and  at 
the  distance  of  5  metres  from  the  mirror.     What  are  the  position  and  size  of  the 
image  ?   A  small  plane  reflector  is  then  placed  at  the  principal  focus  of  the  spherical 
mirror,  at  an  inclination  of  45°  to  the  principal  axis,  its  polished  side  being  next 
the  mirror.     What  will  be  the  new  position  of  the  image  ? 


1142  EXAMPLES   IN   OPTICS. 


REFRACTION. 

(The  index  of  refraction  of  glass  is  to  be  taken  as  I}-,  except  where  otherwise  specified,  and 
the  index  of  refraction  of  water  as  f ). 

52.  The  sine  of  45°  is  V  |,  or  707  nearly.     Hence,  determine  whether  a  ray 
incident  in  water  at  an  angle  of  45°  with  the  surface  will  emerge  or  will  be 
reflected ;  and  determine  the  same  question  for  a  ray  in  glass. 

53.  If  the  index  of  refraction  from  air  into  crown-glass  be  1J,  and  from  air 
into  flint-glass  If,  find  the  index  of  refraction  from  crown-glass  into  flint-glass. 

54.  The  index  of  refraction  from  water  into  oil  of  turpentine  is  I'll;  find  the 
index  of  refraction  from  air  into  oil  of  turpentine. 

55.  The  index  of  refraction  for  a  certain  glass  prism  is  1'6,  and  the  angle  of 
the  prism  is  10°.     Find  approximately  the  deviation  of  a  ray  refracted  through  it 
nearly  symmetrically. 

56.  A  ray  of  light  falls  perpendicularly  on  the  surface  of  an  equilateral  prism 
of  glass  with  a  refracting  angle  of  60°.     What  will  be  the  deviation  produced  by 
the  prism?     Index  of  refraction  of  glass  1'5. 

57.  A  speck  in  the  interior  of  a  piece  of  plate-glass  appears  to  an  observer 
looking  normally  into  the  glass  to  be  2  mm.  from  the  near  surface.     What  is  its 
real  distance  ? 

58.  The  rays  of  a  vertical  sun  are  brought  to  a  focus  by  a  lens  at  a  distance  of 
1  ft.  from  the  lens.    If  the  lens  is  held  just  above  a  smooth  and  deep  pool  of  water, 
at  what  depth  in  the  water  will  the  rays  come  to  a  focus  ? 

59.  A  mass  of  glass  is  bounded  by  a  convex  surface,  and  parallel  rays  incident 
nearly  normally  on  this  surface  come  to  a  focus  in  the  interior  of  the  glass  at  a 
distance  a.     Find  the  focal  length  of  a  plano-convex  lens  of  the  same  convexity, 
supposing  the  rays  to  be  incident  on  the  convex  side. 

60.  Show  that  the  deviation  of  a  ray  going  through  an  air-prism  in  water  is 
towards  the  edge  of  the  prism. 

LENSES,  &c. 

61.  Compare  the  focal  lengths  of  two  lenses  of  the  same  size  and  shape,  one 
of  glass  and  the  other  of  diamond,  their  indices  of  refraction  being  respectively 
1-6  and  2'6. 

62.  If  the  index  of  refraction  of  glass  be  f ,  show  that  the  focal  length  of  an 
equi-convex  glass  lens  is  the  same  as  the  radius  of  curvature  of  either  face. 

63.  The  focal  length  of  a  convex  lens  is  1  ft.     Find  the  positions  of  the  image 
of  a  small  object  when  the  distances  of  the  object  from  the  lens  are  respectively 
20  ft.,  2  ft.,  and  l£  ft.     Are  the  images  real  or  virtual? 

64.  When  the  distances  of  the  object  from  the  lens  in  last  question  are  respec- 
tively 11  in.,  10  in.,  and  1  in.,  find  the  distances  of  the  image.     Are  the  images 
real  or  virtual? 

65.  Calling  the  diameter  of  the  object  unity,  find  the  diameter  of  the  image 
in  the  six  cases  of  questions  63,  64,  taken  in  order. 

66.  Show  that,  when  the  distance  of  an  object  from  a  convex  lens  is  double 
the  focal  length,  the  image  is  at  the  same  distance  on  the  other  side. 

67.  The  object  is  6  ft.  on  one  side  of  a  lens,  and  the  image  is  1  ft.  on  the  other 
side.     What  is  the  focal  length  of  the  lens? 


EXAMPLES   IN   OPTICS.  1143 

68.  The  object  is  3  in.  from  a  lens,  and  its  image  is  18  in.  from  the  lens  on  the 
same  side.     Is  the  lens  convex  or  concave,  and  what  is  its  focal  length? 

69.  The  object  is  12  ft.  from  a  lens,  and  the  image  1  ft.  from  the  lens  on  the 
same  side.     Find  the  focal  length,  and  determine  whether  the  lens  is  convex  or 
concave. 

70.  A  person  who  sees  best  at  the  distance  of  3  ft.,  employs  convex  spectacles 
with  a  focal  length  of  1  ft.     At  what  distance  should  he  hold  a  book,  to  read  it 
with  the  aid  of  these  spectacles? 

71.  A  person  reads  a  book  at  the  distance  of  1  ft.  with  the  aid  of  concave 
spectacles  of  6  in.  focal  length.    At  what  distance  is  the  image  which  he  sees'? 

72.  A  pencil  of  parallel  rays  fall  upon  a  sphere  of   glass  of  1  inch  radius. 
Find  the  principal  focus  of  rays  near  the  axis,  the  index  of  refraction  of  glass 
being  1-5. 

73.  What  is  the  focal  length  of  a  double-convex  lens  of  diamond,  the  radius  of 
curvature  of  each  of  its  faces  being  4  millimetres?     Index  of  refraction  2'5. 

74.  An  object  8  centimetres  high  is  placed  at  1  metre  distance  on  the  axis  of 
an  equi-convex  lens  of  crown-glass  of  index  1*5,  the  radius  of  curvature  of  its  faces 
being  0'4  m.     Find  the  size  and  position  of  the  image. 

75.  Two  converging  lenses,  with  a  common  focal  length  of  0'05  m.,  are  at  a 
distance  of  0'05  in.  apart,  and  their  axes  coincide.     What  image  will  this  system 
give  of  a  circle  O'Ol  m.  in  diameter,  placed  at  a  distance  of  '1  m.  on  the  prolonga- 
tion of  the  common  axis? 

76.  Show  that  if  F  denote  the  focal  length  of  a  combination  of  two  lenses  in 
contact,  their  thicknesses  being  neglected,  we  have 


/i  and/2  denoting  the  focal  lengths  of  the  two  lenses. 

77.  What  is  the  focal  length  of  a  lens  composed  of  a  convex  lens  of  2  in.  focal 
length,  cemented  to  a  concave  lens  of  9  in.  focal  length? 

78.  Apply  the  formulae  of  §  1015  to  find  the  focal  length  of  a  lens,  the  thick- 
ness being  neglected. 

79.  The  objective  of  a  telescope  has  a  focal  length  of  20  ft.     What  will  be  the 
magnifying  power  when  an  eye-piece  of  half -inch  focus  is  used? 

80.  A  sphere  of  glass  of  index  1'5  lying  upon  a  horizontal  plane  receives  the 
sun's  rays.    What  must  be  the  height  of  the  sun  above  the  horizon  that  the  prin- 
cipal focus  of  the  sphere  may  be  in  this  horizontal  plane? 

81.  A  small  plane  mirror  is  placed  exactly  at  the  principal  focus  of  a  telescope, 
nearly  perpendicular  to  its  axis,  and   the    telescope  is  directed  approximately 
to  a  distant  luminous  object.     Show  that  the  rays  reflected  at  the  mirror  will, 
after  repassing  the  object  glass,  return  in  the  exact  direction  from  which  they 
came,  in  spite  of  the  small  errors  of  adjustment  of  the  mirror  and  telescope. 

82.  An  eye  is  placed  close  to  the  surface  of  a  large  sphere  of  glass  (M^f)  which 
is  silvered  at  the  back.    Show  that  the  image  which  the  eye  sees  of  itself  is  three- 
fifths  of  the  natural  size. 

83.  The  refractive  indices  for  the  rays  D  and  F  for  two  specimens  of  glass  are 

Crown-glasa        1-5279         1-5844 

Flint-glass  1-6351         1-6481 


1144:  ANSWERS   TO   EXAMPLES. 

and  an  achromatic  lens  of  20  in.  focal  length  is  to  be  formed  by  their  combination. 
Show  that  if  the  rays  D  and  F  are  brought  to  the  same  focus,  the  sum  of  curvatures 
of  the  two  faces  for  the  crown  lens  must  be  double  that  for  the  flint,  and  the  focal 
lengths  of  the  two  lenses  which  are  combined  will  be  about  7'9  in.  for  the  crown 
and  13-1  in.  for  the  flint. 


ANSWERS  TO  EXAMPLES  IN  ACOUSTICS. 

Ex.  1.  -fa  sec.  50.  Ex.  2.  4.  Ex.  3.  13|  ft.  If  ft.  Ex.  4.  61J  ft.,  6&  ft. 
Ex.  5.  30  ft. 

Ex.  6.  f  xf  =  2.     Ex.  7.  ft  =  f-     Ex-  8-  Do»>  Sol*>  Do»  Mi3>  SoU. 

Ex.  9.  547.     Ex.  10.  440. 

Ex.  11.  825.  Ex.  12.  H  =  '382  sec-  Ex  13-  32  ft->  16  ft->  10i  ft-»  34§>  68i» 
103J.  Ex.  14.  20  ft.,  6§  ft.,  4  ft.;  55,  165,  275.  Ex.  15.  ^  ft.,  ||  ft.  Ex.  16. 
An  open  tube  twice  or  three  times  as  long  will  resound,  because  one  of  its  over- 
tones will  coincide  with  the  note  of  the  fork.  A  stopped  tube  three  times  as  long 
will  resound,  but  a  stopped  tube  twice  as  long  will  not. 

Ex.  17.  4.     Ex.  18.  2,  4,  6.     Ex.  19.  675-672  =  3. 

Ex.  20.  33843,  38850.  Ex.  21.  1110.  Ex.  22.  The  velocity  is  1090  at  32°  and 
1110  at  50°.  Ex.  23.  1-018  metre.  Ex.  24.  4180  ft.  per  second.  Ex.  25.  33732. 
Ex.  26.  1945  vibrations  per  second.  Ex.  27.  364°  C. 

Ex.  28.  37417.  Ex.  29.  567,  1134,  1701.  Ex.  30.  v  =  29040,  t  =  v*m  =  548 1600. 
Ex.  31.  v  =  43560,  t  =  7589900.  Ex.  32.  Unison.  Ex.  33.  Length  of  iron  =  l'68 
times  length  of  platinum. 

Ex.  34.  10  ft.     Ex.  35.  12800  ft.  per  second. 


ANSWERS  TO  EXAMPLES  IN  OPTICS. 

Ex.  36.  2'95  m.  Ex.  37.  1,  5,  and  7  ft.  behind  first  mirror;  2, 4,  and  8  ft.  behind 
second.  Ex.  39.  Side  of  mirror  must  be  ^  of  edge  of  cube. 

Ex.  41.  They  are  the  shadows  of  the  object  and  of  its  image,  cast  by  the  sun's 
image.  The  former  is  due  to  the  intercepting  of  light  after  reflection ;  the  latter 
to  the  intercepting  of  light  before  reflection.  Ex.  42.  The  sun's  image  throws  a 
shadow  of  the  man's  image  on  the  real  arch,  owing  to  his  intercepting  rays  on 
their  way  to  the  water.  Ex.  43.  First  let  the  globe  be  vertically  under  the  flame, 
and  draw  through  the  flame  two  equally  inclined  planes,  touching  the  globe. 
Their  intersections  with  the  table  will  be  parallel  lines  which  will  be  tangents  to 
the  shadow,  and  will  still  remain  tangents  to  it  as  the  globe  is  rolled  between  the 
planes  to  any  distance.  Ex.  44.  216  radii  of  earth;  l£  radii.  Ex.  45.  368  <r=  1156 
metres. 

Ex.  46.  Focal  length  1  ft.;  (a)  5  ft.  in  front  of  mirror;  (b)  1J  ft.  in  front; 
(c)  3  ft.  behind  mirror;  (d)  Ifr  in>  behind.     Ex.  47.  4,  £,  4,  1^. 
'     Ex.  48.  69'6  cm.      Ex.  49.  Distance  §  m.,  height  3  cm.      Ex.  50.  8'73  cm. 
Ex.  51.  Distance  l£  m.,  length  J  dec.,  new  position  |-  m.  laterally  from  focus. 

Ex.  52.  The  ray  in  water  will  emerge,  because  f  is  greater  than  '707 ;  the  ray 


ANSWERS  TO   EXAMPLES.  1145 

in  glass  will  be  totally  reflected,  because  §  is  less  than  '707.  Ex.  53.  }f. 
Ex.  54.  1-48.  Ex.  55.  6°.  Ex.  56.  60°  (by  total  reflection). 

Ex.  57.  3  mm.     Ex.  58.  1  ft.  4  in.    Ex.  59.  §  a. 

Ex.  61.  Focal  length  of  diamond  lens  is  f  of  focal  length  of  glass  lens. 
Ex.  63.  IjV  ft.,  2  ft,  3  ft.  on  other  side  of  lens.  All  real.  Ex.  64.  1 1  ft.,  5  ft., 
T\  ft.  on  same  side  of  lens.  All  virtual.  Ex.  65.  fa  1,  2,  12,  6,  1^-.  Ex.  66.  1^-. 
Ex.  67.  $  ft.  Ex.  68.  3f  in.,  convex.  Ex.  69.  1^  ft.,  concave.  Ex.  70.  9  in. 
Ex.  71.  4  in.  Ex.  72.  1-5  in.  from  centre,  or  '5  in.  from  sphere. 

Ex.  73.  Ij  mm.  Ex.  74.  Distance  §  m.  on  other  side,  height  5J  cm.  Ex.  75.  A 
real  image  '025  m.  beyond  second  lens;  diameter  of  image  '005  m.  Ex.  77.  2f  in. 
Ex.  79.  480.  Ex.  80.  Sine  of  altitude  =  §,  altitude  =  41°  49'.  Ex.  81.  Rays  from 
one  point  of  object  converge  to  one  point  on  mirror,  and  are  reflected  from  this 
point  as  a  new  source.  Hence  by  the  principle  of  conjugate  foci  they  will  return 
to  the  point  whence  they  came.  Ex.  82.  The  first  and  second  images  are  at  dis- 
tances of  ^  and  |  of  radius  from  centre. 

Ex.  83.  The  dispersive  powers  are  as  32:53.  The  focal  lengths  are  to  be 
directly  as  these  numbers,  and  the  difference  of  their  reciprocals  must 'be  fa 


INDEX   TO    PAET    IV. 


Aberration,  astron  3mic.il,  962. 

Collimator  ot  spectroscope,  1069, 

Doppler's  principle,  1077. 
Double  refraction,  1010,  1122. 

c  romatic,  10  o. 

Colour,  1087-1098. 

Duhamel's  vibroscope,  904. 

—  spherical,  978. 
Absorption  and  emission  of  rays, 

—  and  music,  1097. 
—  blindness,  1097. 
—  by  polarized  light,  1124-1133- 

Ear,  according  to  Helmholtz,  943. 
Echo,  887. 

1074. 
Accidental  images,  1096. 

—  cone,  1094. 

Edison's  phonograph,  939. 

—  equations,  1091. 

Elementary  tones,  945. 

Acoustic  pendulum,  892. 
/Ether,  luminiferous,  947. 
Air,  vibration  of,  868. 

—  mixture  of,  1089-1095. 
—  of  powders,  1088. 
—  of  thin  films,  1118. 

Ellipsoid,  1131- 
Elliptic  polarization,  1124. 
Energy    of   sonorous    vibrations, 

"     Air/s  apparatus  for  law  of  sines, 

Domma,  901. 
Complementary  colours,  1095. 

879. 
Extraordinary  index,  1012,  1123. 

Amplitude  of  vibration,  866. 

Concave  lenses,  1023. 

—  rays,  1012,  1123. 

Analyser,  1119. 
Anamorphosis,  990. 
Apertures  form  images,  950. 

—  mirrors,  977. 
Concord,  941. 
Cone  of  colour,  1094. 

Eye,  1031. 
Eye-pieces,  1083. 

Artificial  horizon,  968. 
Astronomical     refraction,      1003, 

Conjugate  foci,  978,  1016. 
Constitution  of  compound  vibra- 

Field of  view,  1056. 
Films,  colours  of,  1118. 

1108. 
Astronomical  telescope,  1043. 
Atmospheric  refraction,  1103. 
Axes,    optic,    in    crystals,    ion, 
1122,  1131. 

Basilar  membrane  of  ear,  943. 

tions,  934. 
Construction  for  image,  982. 
Convex  mirrors,  989,  1087*. 
Cornu  on  velocity  of  light,  957 
Critical  angle,  996. 
Cross  and  rings,  1127. 
Cross-wires    of    telescope,    1022, 
1057. 

Fizeau  on  velocity  of  light,  955. 
Flames,  manometric,  926,  937. 
—  singing,  869. 
Flue-pipe,  917. 
Fluorescence,  1066. 
Flute  mouthpiece,  917. 
Focal  length,  980. 

Beats,  892,  942. 
Bellows  of  organ,  918. 
Bells,  vibration  of,  867,  915. 

Crystals,  optical  classification  of, 
1130. 
Curvature  of  rays  in  air,  1106. 

Focal  lines,  987. 
Foci,  conjugate,  978,  1016. 
—  explained     by    wave    theory, 

Bernoulli's  laws,  919. 
Biaxal  crystals,  1130. 
Binocular  vision,  1033. 

—  of  sound  rays,  1  1  10. 
Cylindric  mirror,  990. 

1104. 
—  primary  and  secondary,  986. 
—  principal,  978,  980,  1014. 

Blackburn's  pendulum,  932,  1125. 

Dark    ends    of  spectrum,    1064- 

Focometer,  1024. 

Block  pipe,  917. 
Brightness,  1050-1056. 
—  intrinsic  and  effective,  1051. 

1067. 
—  lines  in  spectrum,  1064. 
Depolarization,  see  Elliptic  polari- 

Focus,  978. 
Foucault's  experiments  on  velocity 
oflight,  957,  1103. 

—  of  spectra,  1078. 
Bright  spot  behind  eyepiece,  1045. 
Buys  Ballot's  experiment  on  sound, 
907. 

Deviation,  constructions  for,  1008, 
1009. 
—  by  rotation  of  mirror,  976. 

Fourier's  theorem,  934. 
Fraunhofer's  lines,  1064. 
Free  reed,  925. 
Frequencies    of    red    and    violet 

Cngniard  de  Latour's  siren,  902. 

—  minimum,  1008,  1009. 
Difference-tones,  944. 

vibrations,  848. 
Frequency,  896. 

Camera  lucida,  999. 

JJinraction,  noo. 

Fresnel's  rhomb,  1134. 

—  obscura,  1027. 

—  by  grating,  1112,  mo. 

—  wave-surface,  1130. 

—  photographic,  1028. 

—  fringes,  i  ii  i. 

Fringes,  diffraction,  nil. 

Cassegranian  telescope,  1050. 
Caustics,  986,  1000,  1104. 
Centre  of  lens,  1015. 

—  spectrum,  1116. 
Direction  of  vibration  in  polarized 
light,  1136. 

Galilean  telescope,  1047. 

Character  of  a  musical  note,  897, 
933- 
Chemical  harmonica,  869. 
Chladni's  figures,  915. 
Chromatic  aberration,  1080, 

Discord,  941. 
Dispersion,  chromatic,  1059. 
in  spectroscope,  1079. 
Displacement  of  spectral  lines  by 
motion,  1077. 

Gamut,  898. 
Gases,  veloc.  of  sound  in.  883,  024. 
Glass,  strained,  shows  colour,  1132 
Goniometers,  1086. 
Gratings  for  diffraction,  1113- 

Chromosphere,  1075. 
Circular  polarization,  1126,  1134. 
Colladon's  experiment  at  Lake  of 

Dissipation  of  sonorous  energy, 
879. 
Distance,   adaptation  of  eye  to, 

—  concave,  *ni9- 
—  photographic,  1113- 
—  reflection,  1116. 

r<»n«v4     Rfti     run 

1032. 

—  retardation,  1116. 

Oeneva,  003,  949- 
Collimation,  line  of,  1058,  1088*. 

—  judgment  of,  1034. 

Gregorian  telescope,  1049. 

INDEX  TO   PART  IV. 


1147 


Hadley's  sextant,  976. 
Harmonics,   912,   934,   it*  Over- 

lanometric  flames,  926,  937. 
.larch  of  conjugate  foci,  981,  1020. 
>1  ax  well's  colour-box,  1092. 

'ipes,  overtones  of,  919. 
'itch,  896. 
—  modified  by  motion,  906. 

__  ..*       '*      - 

klembrana  basilaris,  943. 

—  standards  of,  900. 

Helmholtz  on  colour,  1089,  1091, 

i_nfi 

lica  plates  for  circular  polariza- 
tion, 1134* 

Mane  mirrors,  970. 
'lane  of  polarization,  1121,  1136. 

1O9O. 

—  resonators,  936. 

Michelson  on  velocity  of  light,  960. 

Mates,  refraction  through,  1001. 

—  theory  of  dissonance,  942. 
Herschelian  telescope,  1048. 

Micrometers,  1058. 
Microscope,  compound,  1041. 

—  superposed,  1003. 
—  vibration  of,  867,  915. 

Huygens'  construction  for  wave- 

—  electric,  1030, 

'olarization,  1119. 

front,  IIOI. 

—  simple,  1039. 

—  by  absorption,  1119. 

—  principle,  1099. 

—  solar,  1029. 
ilinor  scale,  901. 

—  by  double  refraction,  1122, 
—  by  reflection  and  transmission. 

Iceland-spar,  1010,  1122. 

Minimum    deviation    by    prism. 

1120. 

Images,  971. 

1008,  1009. 

—  circular,  1126,  1134. 

—  accidental,  1096. 

ili  rage,  1108. 

—  elliptic,  1124. 

—  brightness  of,  1055. 

Mirrors,  970. 

—  of  dark  rays,  1137. 

—  formed  by  lenses,  1021. 

—concave,  977;  convex,  989,  1087.* 

—  plane  of,  1121,  1136. 

—  formed  by  small  holes,  950. 
—  in  mid  air,  985. 
—  on  screen,  984,  1055. 
—  size  of,  982,  1021. 

—  cylindric,  990. 
—parabolic,  978;  plane,  970. 
Mixture  of  colours,  1089-1095. 
Monochord,  see  Sonometer. 

'olarizer,  1119. 
'nmary  colour-sensations,  1095. 
—  and  secondary  foci,  986. 
Principal  focus,  978,  980,  1014. 

Index  of  refraction,  996,  iioa. 

VIonochromatic  light,  1078,  1128. 

Principle  of  Huygens,  1099. 

table  of,  996. 

Mouth-pieces  of  organ-pipes,  917, 

?rism  in  optics,  1004-1010. 
—  Nicol'sand  Foucault's,  1123. 

Interference  of  sounds,  889-893 

Multiple  images,  973,  1002. 

Projection  by  lenses,  1029. 

Intervals,  musical,  898. 
Invisible  parts  of  spectrum,  1064. 

Jupiter's  satellites,  eclipses  of,  960. 

Musical  sound,  870. 

Newcomb  on  velocity  of  light,  960. 
Newtonian  telescope,  1049. 

Propagation  of  light,  1099. 
—  of  sound,  872-879,  893-895. 
Pure  spectrum,  1062. 
Purity     numerically     measured, 

Kaleidoscope,  974. 

Newton's  rings,  1118. 
—spectrum  experiment,  1060. 

1079. 
Pythagorean  scale,  901. 

Kerr's  magneto-optic  effects,  1134. 
Konig's  manometric  flames,  926, 
937- 

—theory  of  refraction,  1102. 
Nicol's  prism,  1123. 
Nodal  lines  on  plate,  868. 

Quarter-wave  plates,  1134. 
Quartz  rotates  plane  of  polariza- 

Lantern, magic,  1030. 

—  points  of  lens,  1087*. 
Nodes  and  antinodes  in  air,  891. 

tion,  1132. 
—  transparent  to  ultra-violet  rays, 

Laplace's    correction    of    sound- 

in  pipes,  920. 

1067. 

velocity,  883. 

Noise  and  musical  sound.  870. 

Laryngoscope,  991. 
Least  time,  principle  of,  1103. 

Ohm  on  elementary  tones,  945. 

Rainbow,  1083. 
Rankine  on  propagation  of  sound, 

Lenses,  1013. 
—  centre  of  lens,  1015. 

Opera-glass,  1048. 
Ophthalmoscope,  991. 

Recomposition  of  white  light,  1067. 

—  concave,  1023. 

Optical  centre  of  lens,  1015. 

Rectilinear  propagation,  948,  iioo. 

—  formulae  for,  1017. 

—  examination  of  vibrations,  927- 

Reed-pipes,  924. 

Levelling,  corrections  in,  1105. 

933- 

Reflection  of  light,  967. 

Light,  947-1137. 

Optic  axes  in  biaxal  crystals,  1131 

irregular,  969,  1088. 

Limma,  899. 

—  axis  in  uniaxal  crystals,  ion 

total,  998. 

Linear  dimensions,  in  sound,  916 

1  1  22. 

—  of  sound,  886. 

919. 

Ordinary  and  extraordinary  image 

Refraction,  992. 

Line  of  collimation,  1058. 

1012,  1123. 

—  astronomical,  1003,  1108. 

Lines,  Fraurihofer's,  1064. 
Lissajous"  curves,  929. 

Organ-pipes,  917-925. 
effect    of   temperature  on 

—  at  plane  surface,  1000. 
—  at  spherical  surface,  1025. 

Log  of  wood,  propagation  of  sounc 
through,  879. 
Longitudinal   and   transverse  vi 

925. 
overtones  of,  919. 
Overtones,  912,  914,  919. 

—  atmospheric,  1105. 
—  double,  1010,  1122. 
—  Newtonian  explanation  of,  1  102. 

brations,  908. 

—  of  sound,  887,  mo. 

—  vibrations  of  rods  and  strings 

Parabolic  mirrors,  978. 

—  table  of  indices  of,  996. 

923. 

Parallel  mirrors,  972. 

—  undulatory  explanation  of,  1102. 

Looking-glasses,  9701 

Pencil,  979. 

Regnault  on  velocity  of  sound, 

Louduess,  896. 

Pendulum,  acoustic,  893 

878. 

Luminiferous  zther,  947. 

Penumbra,  954. 

Resonance,  913. 

Lycopodium  on  vibrating  plate 

Pepper's  ghost,  975. 

Resonators,  936. 

868. 

Period,  871. 

Resultant  tones,  944. 

Phantom  bouquet,  984. 

Reversal  of  bright  lines,  1073. 

Magic-lantern,  1030. 

Phonautograph,  905. 

Rhomb,  Fresnel's,  1134. 

Magneto-optic  rotation,  1133. 

Phonograph,  Edison's,  939. 

Rings  by  polarized  light,  1127. 

Magnification,  1036. 
—  by  lens,  1040. 

Phosphorescence,  1065. 
Phosphoroscope,  1065. 

—  Newton's,  in  8. 
Rock-salt,  its  diathermancy,  1064 

—  by  microscope,  1041. 

Photography,  1028. 

Rods,  vibrations  of,  923. 

—  by  telescope,  1044,  1046. 

Photometers,  963-966. 

Rotation  of  mirror,  976. 

Malus'  polariscope,  mo. 

Photosphere,  1075. 

—  of  plane  of  polarization,  1132. 

1148 


INDEX  TO  PART  IV. 


Saccharine  solutions,  by  polarized 

Speculum-metal,  970. 

Trumpet,  speaking  and  hearing, 

light,  1132. 

Sphere,  refraction  through;  1026. 

8b8. 

Scale,  musical,  898. 

Spherical  mirrors,  977-990. 

Tubes,     propagation     of    sound 

Scattered  light,  969,  1088. 

—  aberration,  978. 

through,  878. 

Screen,  image  on,  984,  1055. 

Spring,  vibration  of,  865. 

Tuning-fork,  916. 

Secondary  axis,  932,  978,  1016. 
Segmental  vibration,  867,  912,  914, 
920. 

Squares,  inverse,  879. 
Stars,  brightness  of,  1054. 
—  motion  of,  1077. 

Umbra  and  penumbra,  954. 
Unannealed    glass,   by  polarized 

Selective  emission  and  absorption, 
1074. 
Selenite  by  polarized  light,  1124. 
Semitone,  900,  901. 
Sextant,  976. 

—  spectra  of,  1077. 
Stationary  undulations,  891,  895, 
921. 
Stereoscope,  1034. 
Stethoscope,  872. 

light,  1132. 
Undulation,  definition  of,  877. 
—  nature  of,  875,  1099. 
—  stationary,  891,  895,  921. 
Uniaxal  crystals,  1010,  1122,  1131. 

Shadows,  952. 
-  for  sound,  948. 

Stops  of  organs,  935. 
Strained  glass,  by  polarized  light, 

Velocity  of  light,  955-963. 
—  of  sound  in  air,  880-883. 

Simple   tones   arise   from   simple 

1132. 

vibrations,  945. 

Striking  reed,  925. 

in  liquids,  883. 

Sines,  law  of,  994,  1102. 

Stringed  instruments,  914,  915. 

in  solids,  844,  884. 

Singing  flames,  869. 

Strings,    vibrations    of,    908-915, 

mathematically     investi- 

Siren, 902. 
Sirius,  motion  of,  1077. 
Small  holes  form  images,  950. 
Sodium  line,  1073. 
Solar  microscope,  1029. 

923- 
Successive  reflections,  972. 
Summation-tones,  944. 
Sun,  atmosphere  of,  1074. 
—  distance  of,  961. 

gated,  893,  894. 
Vibrations  of  ordinary  light,  1136. 
—  of  plane  polarized  light,  1136. 
—  single  and  double,  865. 
—  transverse    and     longitudinal, 

—  spectrum,  1060-1065. 

—  see  Solar. 

875,  877    908. 

Sondhaus'  experiment,  887. 

Swan  on  the  sodium  line,  1073. 

Vibroscope,  904. 

Sonometer,  911. 

Synthesis  of  vowel  sound,  938. 

Virtual  images,  987,  988,  1023. 

Sound,  865-946. 

T     •  •, 

Vision,  1032. 

—  propagation  of,  872,  893. 

Telescopes,  1043-1058. 

Visual  angle,  1036. 

—  reflection  of,  886. 

Telespectroscope,  1075. 

'  *•*'' 

—  refraction  of,  887,  uro. 

Temperament,  899. 

Water,  velocity  of  sound  in,  883. 

—  shadows  in  water,  949. 

Terrestrial  refraction,  1105. 

Wave-front,  1099. 

—  curved  rays  of,  mo. 
—  vehicle  of,  871. 

Thin  films,  colours  of,  1118. 
Timbre,  897,  933. 

Wave-lengths  of  light,  1117. 
of  sound,  874,  897. 

Sounding-boards,  914. 

Tones,  major  and  minor,  899. 

relation  of,  to  velocity  and 

Speaking-trumpet,  888. 

—  resultant,  944. 

frequency,  874,  897,  948. 

Spectacles,  1037. 

Tonometer,  905. 

Wave-surface,  1123,  1130. 

Spectra,  1072. 

Total  reflection,  998. 

—  theory  of  light,  1099. 

—  brightness  and  purity  of,  1078. 

Tourmalines,  1119,  1123. 

Wertheim's  experiments  on  velo- 

— by  diffraction,  1112. 

Transmission  of  sound,  872,  893. 

city  of  sound,  885,  924. 

Spectroscope,  1069. 

Transverse  and  longitudinal  vibra- 

Wind, effect  of,  on  sound,  mo. 

Spectrum  analysis,  1072. 

tions,  875,  877,  908. 

chest,  918. 

Specula,  silvered,  1050. 

Trevelyan  experiment,  869. 

—  instruments,  925. 

THE  ENU 


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